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Properties

Label 2-431-431.101-c2-0-50
Degree 22
Conductor 431431
Sign −0.202+0.979i-0.202 + 0.979i
Analytic cond. 11.743811.7438
Root an. cond. 3.426933.42693
Motivic weight 22
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.834 − 1.62i)2-s + (−0.555 − 5.05i)3-s + (0.389 + 0.544i)4-s + (6.62 + 3.10i)5-s + (−8.66 − 3.31i)6-s + (1.68 + 7.57i)7-s + (8.43 − 1.24i)8-s + (−16.4 + 3.65i)9-s + (10.5 − 8.15i)10-s + (4.07 − 2.28i)11-s + (2.53 − 2.27i)12-s + (−3.71 + 1.90i)13-s + (13.7 + 3.58i)14-s + (12.0 − 35.1i)15-s + (4.15 − 12.1i)16-s + (−10.2 − 18.3i)17-s + ⋯
L(s)  = 1  + (0.417 − 0.811i)2-s + (−0.185 − 1.68i)3-s + (0.0974 + 0.136i)4-s + (1.32 + 0.621i)5-s + (−1.44 − 0.552i)6-s + (0.241 + 1.08i)7-s + (1.05 − 0.155i)8-s + (−1.82 + 0.406i)9-s + (1.05 − 0.815i)10-s + (0.370 − 0.208i)11-s + (0.211 − 0.189i)12-s + (−0.285 + 0.146i)13-s + (0.979 + 0.256i)14-s + (0.800 − 2.34i)15-s + (0.259 − 0.761i)16-s + (−0.605 − 1.07i)17-s + ⋯

Functional equation

Λ(s)=(431s/2 ΓC(s) L(s)=((−0.202+0.979i) Λ‾(3−s)\begin{aligned}\Lambda(s)=\mathstrut & 431 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.202 + 0.979i)\, \overline{\Lambda}(3-s) \end{aligned}
Λ(s)=(431s/2 ΓC(s+1) L(s)=((−0.202+0.979i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 431 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.202 + 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 431431
Sign: −0.202+0.979i-0.202 + 0.979i
Analytic conductor: 11.743811.7438
Root analytic conductor: 3.426933.42693
Motivic weight: 22
Rational: no
Arithmetic: yes
Character: χ431(101,⋅)\chi_{431} (101, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 431, ( :1), −0.202+0.979i)(2,\ 431,\ (\ :1),\ -0.202 + 0.979i)

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 1.83490−2.25344i1.83490 - 2.25344i
L(12)L(\frac12) ≈\approx 1.83490−2.25344i1.83490 - 2.25344i
L(2)L(2) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad431 1+(429.+38.3i)T 1 + (429. + 38.3i)T
good2 1+(−0.834+1.62i)T+(−2.32−3.25i)T2 1 + (-0.834 + 1.62i)T + (-2.32 - 3.25i)T^{2}
3 1+(0.555+5.05i)T+(−8.78+1.95i)T2 1 + (0.555 + 5.05i)T + (-8.78 + 1.95i)T^{2}
5 1+(−6.62−3.10i)T+(15.9+19.2i)T2 1 + (-6.62 - 3.10i)T + (15.9 + 19.2i)T^{2}
7 1+(−1.68−7.57i)T+(−44.3+20.7i)T2 1 + (-1.68 - 7.57i)T + (-44.3 + 20.7i)T^{2}
11 1+(−4.07+2.28i)T+(63.0−103.i)T2 1 + (-4.07 + 2.28i)T + (63.0 - 103. i)T^{2}
13 1+(3.71−1.90i)T+(98.3−137.i)T2 1 + (3.71 - 1.90i)T + (98.3 - 137. i)T^{2}
17 1+(10.2+18.3i)T+(−150.+246.i)T2 1 + (10.2 + 18.3i)T + (-150. + 246. i)T^{2}
19 1+(−26.8+12.5i)T+(230.−277.i)T2 1 + (-26.8 + 12.5i)T + (230. - 277. i)T^{2}
23 1+(1.65+1.71i)T+(−19.3+528.i)T2 1 + (1.65 + 1.71i)T + (-19.3 + 528. i)T^{2}
29 1+(−6.96+9.73i)T+(−271.−795.i)T2 1 + (-6.96 + 9.73i)T + (-271. - 795. i)T^{2}
31 1+(5.81+15.1i)T+(−715.+641.i)T2 1 + (5.81 + 15.1i)T + (-715. + 641. i)T^{2}
37 1+(16.6+4.36i)T+(1.19e3+670.i)T2 1 + (16.6 + 4.36i)T + (1.19e3 + 670. i)T^{2}
41 1+(9.02−26.4i)T+(−1.33e3−1.02e3i)T2 1 + (9.02 - 26.4i)T + (-1.33e3 - 1.02e3i)T^{2}
43 1+(11.9−15.4i)T+(−467.−1.78e3i)T2 1 + (11.9 - 15.4i)T + (-467. - 1.78e3i)T^{2}
47 1+(4.26−58.2i)T+(−2.18e3−321.i)T2 1 + (4.26 - 58.2i)T + (-2.18e3 - 321. i)T^{2}
53 1+(−45.6+3.33i)T+(2.77e3−408.i)T2 1 + (-45.6 + 3.33i)T + (2.77e3 - 408. i)T^{2}
59 1+(23.4−45.6i)T+(−2.02e3−2.83e3i)T2 1 + (23.4 - 45.6i)T + (-2.02e3 - 2.83e3i)T^{2}
61 1+(42.7+100.i)T+(−2.58e3+2.67e3i)T2 1 + (42.7 + 100. i)T + (-2.58e3 + 2.67e3i)T^{2}
67 1+(102.+26.8i)T+(3.91e3+2.19e3i)T2 1 + (102. + 26.8i)T + (3.91e3 + 2.19e3i)T^{2}
71 1+(8.46−115.i)T+(−4.98e3−733.i)T2 1 + (8.46 - 115. i)T + (-4.98e3 - 733. i)T^{2}
73 1+(−68.0−102.i)T+(−2.08e3+4.90e3i)T2 1 + (-68.0 - 102. i)T + (-2.08e3 + 4.90e3i)T^{2}
79 1+(−0.0698+0.232i)T+(−5.20e3−3.44e3i)T2 1 + (-0.0698 + 0.232i)T + (-5.20e3 - 3.44e3i)T^{2}
83 1+(61.2−6.73i)T+(6.72e3−1.49e3i)T2 1 + (61.2 - 6.73i)T + (6.72e3 - 1.49e3i)T^{2}
89 1+(48.4−8.95i)T+(7.39e3−2.82e3i)T2 1 + (48.4 - 8.95i)T + (7.39e3 - 2.82e3i)T^{2}
97 1+(−42.0+12.6i)T+(7.84e3−5.19e3i)T2 1 + (-42.0 + 12.6i)T + (7.84e3 - 5.19e3i)T^{2}
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   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−11.31162445908612108978269784039, −9.877872035267367662087041050008, −8.938943309295486892860562217828, −7.64578771451439856780742070732, −6.88441287815151932080996932157, −6.05948536553145036947774555152, −5.08175980094882846412363444027, −2.82165042885668209236122151652, −2.39905451297708194830211085888, −1.35892269442789498086938730437, 1.56537090863694857887723635179, 3.72161178495491579433424815731, 4.69549536126449970307503666084, 5.34790858309710404545775134973, 6.11133561502563668333538864617, 7.30030093822099714110119102155, 8.675195508182321560236011677896, 9.588431767533309353676514376469, 10.43243539175299955314290584816, 10.55936861147465051880249087484

Graph of the ZZ-function along the critical line