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Properties

Label 2-637-91.51-c1-0-25
Degree $2$
Conductor $637$
Sign $-0.855 + 0.518i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.855 + 0.518i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.855 + 0.518i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $-0.855 + 0.518i$
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: \((2,\ 637,\ (\ :1/2),\ -0.855 + 0.518i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0352432 - 0.126137i\)
\(L(\frac12)\) \(\approx\) \(0.0352432 - 0.126137i\)
\(L(\frac{3}{2})\) 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

−10.04190218528312069485855013859, −9.103952320186661034107969121993, −8.914729499159470788759718524985, −8.225796063316820588673286843671, −7.02197473604083284215472965704, −5.03791807145877929341731510052, −4.52902368153434017037729271916, −3.52619441812530238168188435003, −2.25085248763119429025874432587, −0.085395496088323932036127400653, 1.67830343170457241254821187881, 3.16042514878099686766922819577, 4.05490805956521754279191936260, 6.29093355918243117619486312964, 6.86382764833778120405177061399, 7.71128828677160508374612866576, 8.008963709710055093018707208089, 8.770001908049390720027989911980, 9.744847265946908769051815768476, 10.90573028409457349771791556282

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