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Properties

Label 2-637-91.51-c1-0-40
Degree $2$
Conductor $637$
Sign $-0.913 - 0.406i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.913 - 0.406i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.913 - 0.406i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $-0.913 - 0.406i$
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: \((2,\ 637,\ (\ :1/2),\ -0.913 - 0.406i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0957815 + 0.451413i\)
\(L(\frac12)\) \(\approx\) \(0.0957815 + 0.451413i\)
\(L(\frac{3}{2})\) 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

−10.47022810858529252963656250992, −8.788649848658990341464436423168, −8.018751063591000364260420708647, −7.09134010259008122000131907544, −6.52798387521593966063232480028, −5.56299503773865759270020033550, −4.75235815185561140393158077922, −3.73685493578888187235992974350, −1.67698707537374106372512853292, −0.20693679086613919407483073510, 3.11091330936756330290561809307, 3.54159006384376207963266477907, 4.47526550219641483524758358435, 5.27934113606144685837312587111, 6.09162037814198044246297775181, 7.63570874994043406782725868565, 8.466234573977176287230138246122, 9.729500065118978954640954849481, 10.64545167078199904008028420187, 11.18460231621733827147746241308

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