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Properties

Label 2-637-91.51-c1-0-10
Degree $2$
Conductor $637$
Sign $-0.248 + 0.968i$
Analytic cond. $5.08647$
Root an. cond. $2.25532$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.31 − 0.761i)2-s + (−1.49 − 2.59i)3-s + (0.160 + 0.277i)4-s + (2.55 + 1.47i)5-s + 4.56i·6-s + 2.55i·8-s + (−2.98 + 5.17i)9-s + (−2.24 − 3.88i)10-s + (2.05 − 1.18i)11-s + (0.480 − 0.831i)12-s + (2.32 − 2.75i)13-s − 8.82i·15-s + (2.26 − 3.93i)16-s + (2.68 + 4.65i)17-s + (7.87 − 4.54i)18-s + (4.63 + 2.67i)19-s + ⋯
L(s)  = 1  + (−0.932 − 0.538i)2-s + (−0.864 − 1.49i)3-s + (0.0801 + 0.138i)4-s + (1.14 + 0.658i)5-s + 1.86i·6-s + 0.904i·8-s + (−0.995 + 1.72i)9-s + (−0.709 − 1.22i)10-s + (0.619 − 0.357i)11-s + (0.138 − 0.240i)12-s + (0.645 − 0.763i)13-s − 2.27i·15-s + (0.567 − 0.982i)16-s + (0.651 + 1.12i)17-s + (1.85 − 1.07i)18-s + (1.06 + 0.614i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.248 + 0.968i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.248 + 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $-0.248 + 0.968i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (324, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ -0.248 + 0.968i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.525524 - 0.677057i\)
\(L(\frac12)\) \(\approx\) \(0.525524 - 0.677057i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
13 \( 1 + (-2.32 + 2.75i)T \)
good2 \( 1 + (1.31 + 0.761i)T + (1 + 1.73i)T^{2} \)
3 \( 1 + (1.49 + 2.59i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (-2.55 - 1.47i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (-2.05 + 1.18i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (-2.68 - 4.65i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-4.63 - 2.67i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.39 + 2.41i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 0.585T + 29T^{2} \)
31 \( 1 + (-7.21 + 4.16i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-0.585 - 0.337i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 - 11.2iT - 41T^{2} \)
43 \( 1 + 10.5T + 43T^{2} \)
47 \( 1 + (-0.465 - 0.268i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (3.95 + 6.84i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-5.32 + 3.07i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.39 + 2.41i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.64 + 2.10i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 1.25iT - 71T^{2} \)
73 \( 1 + (9.40 - 5.43i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (3.14 - 5.44i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 1.74iT - 83T^{2} \)
89 \( 1 + (-8.86 - 5.11i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + 9.90iT - 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.23290374829550344187694818314, −9.842894709890034605920075622420, −8.404594872663841222485855427444, −7.901735248279287857905348926539, −6.55879882192035849625703228465, −6.06759400391498384811420571942, −5.33723547986845053254775782914, −2.99259239195127363649354375155, −1.74105136365210773698716683521, −1.01382900580084368348306935751, 1.07966564497352313426924796944, 3.43631428340104486348906946737, 4.61248054949407649887745330824, 5.35363811081244411861859002455, 6.29827914264466570746697009513, 7.21717665391946573696213821899, 8.799425785413722046108349696565, 9.190763780663685489310236596792, 9.788107813170862274417298368115, 10.32832098936478179457460438264

Graph of the $Z$-function along the critical line