↑

Properties

Label 2-14e2-196.103-c1-0-1
Degree 22
Conductor 196196
Sign −0.965−0.260i-0.965 - 0.260i
Analytic cond. 1.565061.56506
Root an. cond. 1.251021.25102
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.933 + 1.06i)2-s + (−1.89 − 1.29i)3-s + (−0.258 − 1.98i)4-s + (0.459 + 0.0344i)5-s + (3.14 − 0.809i)6-s + (0.101 + 2.64i)7-s + (2.34 + 1.57i)8-s + (0.830 + 2.11i)9-s + (−0.464 + 0.455i)10-s + (−5.59 − 2.19i)11-s + (−2.07 + 4.09i)12-s + (−4.33 + 3.45i)13-s + (−2.90 − 2.35i)14-s + (−0.826 − 0.659i)15-s + (−3.86 + 1.02i)16-s + (2.11 + 2.27i)17-s + ⋯
L(s)  = 1  + (−0.659 + 0.751i)2-s + (−1.09 − 0.746i)3-s + (−0.129 − 0.991i)4-s + (0.205 + 0.0153i)5-s + (1.28 − 0.330i)6-s + (0.0385 + 0.999i)7-s + (0.830 + 0.557i)8-s + (0.276 + 0.705i)9-s + (−0.147 + 0.144i)10-s + (−1.68 − 0.661i)11-s + (−0.599 + 1.18i)12-s + (−1.20 + 0.958i)13-s + (−0.776 − 0.630i)14-s + (−0.213 − 0.170i)15-s + (−0.966 + 0.256i)16-s + (0.512 + 0.552i)17-s + ⋯

Functional equation

Λ(s)=(196s/2 ΓC(s) L(s)=((−0.965−0.260i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.965 - 0.260i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(196s/2 ΓC(s+1/2) L(s)=((−0.965−0.260i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.965 - 0.260i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 196196    =    22⋅722^{2} \cdot 7^{2}
Sign: −0.965−0.260i-0.965 - 0.260i
Analytic conductor: 1.565061.56506
Root analytic conductor: 1.251021.25102
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ196(103,⋅)\chi_{196} (103, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 196, ( :1/2), −0.965−0.260i)(2,\ 196,\ (\ :1/2),\ -0.965 - 0.260i)

Particular Values

L(1)L(1) ≈\approx 0.0216920+0.163743i0.0216920 + 0.163743i
L(12)L(\frac12) ≈\approx 0.0216920+0.163743i0.0216920 + 0.163743i
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)
bad2 1+(0.933−1.06i)T 1 + (0.933 - 1.06i)T
7 1+(−0.101−2.64i)T 1 + (-0.101 - 2.64i)T
good3 1+(1.89+1.29i)T+(1.09+2.79i)T2 1 + (1.89 + 1.29i)T + (1.09 + 2.79i)T^{2}
5 1+(−0.459−0.0344i)T+(4.94+0.745i)T2 1 + (-0.459 - 0.0344i)T + (4.94 + 0.745i)T^{2}
11 1+(5.59+2.19i)T+(8.06+7.48i)T2 1 + (5.59 + 2.19i)T + (8.06 + 7.48i)T^{2}
13 1+(4.33−3.45i)T+(2.89−12.6i)T2 1 + (4.33 - 3.45i)T + (2.89 - 12.6i)T^{2}
17 1+(−2.11−2.27i)T+(−1.27+16.9i)T2 1 + (-2.11 - 2.27i)T + (-1.27 + 16.9i)T^{2}
19 1+(3.06−5.30i)T+(−9.5−16.4i)T2 1 + (3.06 - 5.30i)T + (-9.5 - 16.4i)T^{2}
23 1+(−3.35+3.61i)T+(−1.71−22.9i)T2 1 + (-3.35 + 3.61i)T + (-1.71 - 22.9i)T^{2}
29 1+(−0.395−1.73i)T+(−26.1+12.5i)T2 1 + (-0.395 - 1.73i)T + (-26.1 + 12.5i)T^{2}
31 1+(−1.92−3.33i)T+(−15.5+26.8i)T2 1 + (-1.92 - 3.33i)T + (-15.5 + 26.8i)T^{2}
37 1+(2.78+0.859i)T+(30.5+20.8i)T2 1 + (2.78 + 0.859i)T + (30.5 + 20.8i)T^{2}
41 1+(4.16+8.64i)T+(−25.5+32.0i)T2 1 + (4.16 + 8.64i)T + (-25.5 + 32.0i)T^{2}
43 1+(−0.134+0.278i)T+(−26.8−33.6i)T2 1 + (-0.134 + 0.278i)T + (-26.8 - 33.6i)T^{2}
47 1+(−3.46+0.522i)T+(44.9−13.8i)T2 1 + (-3.46 + 0.522i)T + (44.9 - 13.8i)T^{2}
53 1+(0.191−0.0589i)T+(43.7−29.8i)T2 1 + (0.191 - 0.0589i)T + (43.7 - 29.8i)T^{2}
59 1+(−0.0565−0.755i)T+(−58.3+8.79i)T2 1 + (-0.0565 - 0.755i)T + (-58.3 + 8.79i)T^{2}
61 1+(−0.954+3.09i)T+(−50.4−34.3i)T2 1 + (-0.954 + 3.09i)T + (-50.4 - 34.3i)T^{2}
67 1+(−10.0+5.77i)T+(33.5−58.0i)T2 1 + (-10.0 + 5.77i)T + (33.5 - 58.0i)T^{2}
71 1+(10.3+2.35i)T+(63.9+30.8i)T2 1 + (10.3 + 2.35i)T + (63.9 + 30.8i)T^{2}
73 1+(0.0295−0.196i)T+(−69.7−21.5i)T2 1 + (0.0295 - 0.196i)T + (-69.7 - 21.5i)T^{2}
79 1+(−3.05−1.76i)T+(39.5+68.4i)T2 1 + (-3.05 - 1.76i)T + (39.5 + 68.4i)T^{2}
83 1+(3.10−3.88i)T+(−18.4−80.9i)T2 1 + (3.10 - 3.88i)T + (-18.4 - 80.9i)T^{2}
89 1+(8.95−3.51i)T+(65.2−60.5i)T2 1 + (8.95 - 3.51i)T + (65.2 - 60.5i)T^{2}
97 1−4.04iT−97T2 1 - 4.04iT - 97T^{2}
show more
show less
   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

−12.70082929059331725332319971131, −12.05595341153723172398013869990, −10.85661414227293525828335192852, −10.07579559238989790725666320003, −8.736820317896575522408671219192, −7.80611111034398586615495178935, −6.69030381486494357845233978084, −5.74954021809643956768618248515, −5.13377326186017441879429968550, −2.08510148599544413819026188288, 0.19074014110650379077927741108, 2.74172650411681973127063406528, 4.52422932915530641073299258042, 5.27110433439120008627861240285, 7.18373947335078378893266318239, 7.938859949046600845412921104320, 9.784428706987074059578622912570, 10.09072098390664682831260243978, 10.88499486491594896625663638945, 11.69618368042946166938604390280

Graph of the ZZ-function along the critical line