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Properties

Label 2-507-39.32-c1-0-22
Degree 22
Conductor 507507
Sign 0.944−0.328i0.944 - 0.328i
Analytic cond. 4.048414.04841
Root an. cond. 2.012062.01206
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.965 − 0.258i)2-s + (1.72 − 0.158i)3-s + (−0.866 + 0.5i)4-s + (1.41 + 1.41i)5-s + (1.62 − 0.599i)6-s + (0.366 − 1.36i)7-s + (−2.12 + 2.12i)8-s + (2.94 − 0.548i)9-s + (1.73 + 1.00i)10-s + (1.03 + 3.86i)11-s + (−1.41 + i)12-s − 1.41i·14-s + (2.66 + 2.21i)15-s + (−0.500 + 0.866i)16-s + (2.70 − 1.29i)18-s + (−1.36 − 0.366i)19-s + ⋯
L(s)  = 1  + (0.683 − 0.183i)2-s + (0.995 − 0.0917i)3-s + (−0.433 + 0.250i)4-s + (0.632 + 0.632i)5-s + (0.663 − 0.244i)6-s + (0.138 − 0.516i)7-s + (−0.749 + 0.749i)8-s + (0.983 − 0.182i)9-s + (0.547 + 0.316i)10-s + (0.312 + 1.16i)11-s + (−0.408 + 0.288i)12-s − 0.377i·14-s + (0.687 + 0.571i)15-s + (−0.125 + 0.216i)16-s + (0.638 − 0.304i)18-s + (−0.313 − 0.0839i)19-s + ⋯

Functional equation

Λ(s)=(507s/2 ΓC(s) L(s)=((0.944−0.328i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.944 - 0.328i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(507s/2 ΓC(s+1/2) L(s)=((0.944−0.328i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.944 - 0.328i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 507507    =    3⋅1323 \cdot 13^{2}
Sign: 0.944−0.328i0.944 - 0.328i
Analytic conductor: 4.048414.04841
Root analytic conductor: 2.012062.01206
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ507(188,⋅)\chi_{507} (188, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 507, ( :1/2), 0.944−0.328i)(2,\ 507,\ (\ :1/2),\ 0.944 - 0.328i)

Particular Values

L(1)L(1) ≈\approx 2.63305+0.444262i2.63305 + 0.444262i
L(12)L(\frac12) ≈\approx 2.63305+0.444262i2.63305 + 0.444262i
L(32)L(\frac{3}{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)
bad3 1+(−1.72+0.158i)T 1 + (-1.72 + 0.158i)T
13 1 1
good2 1+(−0.965+0.258i)T+(1.73−i)T2 1 + (-0.965 + 0.258i)T + (1.73 - i)T^{2}
5 1+(−1.41−1.41i)T+5iT2 1 + (-1.41 - 1.41i)T + 5iT^{2}
7 1+(−0.366+1.36i)T+(−6.06−3.5i)T2 1 + (-0.366 + 1.36i)T + (-6.06 - 3.5i)T^{2}
11 1+(−1.03−3.86i)T+(−9.52+5.5i)T2 1 + (-1.03 - 3.86i)T + (-9.52 + 5.5i)T^{2}
17 1+(−8.5+14.7i)T2 1 + (-8.5 + 14.7i)T^{2}
19 1+(1.36+0.366i)T+(16.4+9.5i)T2 1 + (1.36 + 0.366i)T + (16.4 + 9.5i)T^{2}
23 1+(−4.24+7.34i)T+(−11.5−19.9i)T2 1 + (-4.24 + 7.34i)T + (-11.5 - 19.9i)T^{2}
29 1+(2.44+1.41i)T+(14.5+25.1i)T2 1 + (2.44 + 1.41i)T + (14.5 + 25.1i)T^{2}
31 1+(5−5i)T−31iT2 1 + (5 - 5i)T - 31iT^{2}
37 1+(1.36−0.366i)T+(32.0−18.5i)T2 1 + (1.36 - 0.366i)T + (32.0 - 18.5i)T^{2}
41 1+(1.93−0.517i)T+(35.5−20.5i)T2 1 + (1.93 - 0.517i)T + (35.5 - 20.5i)T^{2}
43 1+(5.19−3i)T+(21.5−37.2i)T2 1 + (5.19 - 3i)T + (21.5 - 37.2i)T^{2}
47 1+(−2.82+2.82i)T−47iT2 1 + (-2.82 + 2.82i)T - 47iT^{2}
53 1+5.65iT−53T2 1 + 5.65iT - 53T^{2}
59 1+(3.86+1.03i)T+(51.0+29.5i)T2 1 + (3.86 + 1.03i)T + (51.0 + 29.5i)T^{2}
61 1+(4+6.92i)T+(−30.5+52.8i)T2 1 + (4 + 6.92i)T + (-30.5 + 52.8i)T^{2}
67 1+(1.83+6.83i)T+(−58.0+33.5i)T2 1 + (1.83 + 6.83i)T + (-58.0 + 33.5i)T^{2}
71 1+(1.03−3.86i)T+(−61.4−35.5i)T2 1 + (1.03 - 3.86i)T + (-61.4 - 35.5i)T^{2}
73 1+(−1−i)T+73iT2 1 + (-1 - i)T + 73iT^{2}
79 1+10T+79T2 1 + 10T + 79T^{2}
83 1+(−5.65−5.65i)T+83iT2 1 + (-5.65 - 5.65i)T + 83iT^{2}
89 1+(3.62+13.5i)T+(−77.0+44.5i)T2 1 + (3.62 + 13.5i)T + (-77.0 + 44.5i)T^{2}
97 1+(9.56+2.56i)T+(84.0+48.5i)T2 1 + (9.56 + 2.56i)T + (84.0 + 48.5i)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

−10.83368130394384487406292892081, −10.02306641398152984068868578084, −9.181306871112426471185629103552, −8.363366927644870206672710612715, −7.25068256747469242230099165881, −6.49570382668565400055824815438, −4.93189139056625842958961761898, −4.13653072911740632463002757049, −3.04927052367034177292258821531, −2.02140599346058453468075572188, 1.48453424146428443847951494766, 3.12178148521797688312611986393, 4.06075463510290048440173933567, 5.30021512423940767420754584337, 5.83783897561031449915478876226, 7.20480140386322465647028976534, 8.493628987461634775393539960681, 9.120851529853282065557123181277, 9.550438281994422834297846268667, 10.79224929667812405353984417176

Graph of the ZZ-function along the critical line