Properties

Label 2-637-49.37-c1-0-16
Degree $2$
Conductor $637$
Sign $0.570 - 0.821i$
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  + (−1.15 + 0.785i)2-s + (1.86 − 1.73i)3-s + (−0.0200 + 0.0509i)4-s + (−1.72 − 0.532i)5-s + (−0.789 + 3.46i)6-s + (−0.964 + 2.46i)7-s + (−0.637 − 2.79i)8-s + (0.259 − 3.46i)9-s + (2.40 − 0.743i)10-s + (0.185 + 2.48i)11-s + (0.0509 + 0.129i)12-s + (0.900 + 0.433i)13-s + (−0.824 − 3.59i)14-s + (−4.14 + 1.99i)15-s + (2.84 + 2.64i)16-s + (7.11 + 1.07i)17-s + ⋯
L(s)  = 1  + (−0.814 + 0.555i)2-s + (1.07 − 0.999i)3-s + (−0.0100 + 0.0254i)4-s + (−0.772 − 0.238i)5-s + (−0.322 + 1.41i)6-s + (−0.364 + 0.931i)7-s + (−0.225 − 0.987i)8-s + (0.0866 − 1.15i)9-s + (0.762 − 0.235i)10-s + (0.0560 + 0.747i)11-s + (0.0147 + 0.0374i)12-s + (0.249 + 0.120i)13-s + (−0.220 − 0.961i)14-s + (−1.07 + 0.515i)15-s + (0.712 + 0.661i)16-s + (1.72 + 0.260i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.570 - 0.821i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.570 - 0.821i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $0.570 - 0.821i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (625, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ 0.570 - 0.821i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.968672 + 0.506533i\)
\(L(\frac12)\) \(\approx\) \(0.968672 + 0.506533i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (0.964 - 2.46i)T \)
13 \( 1 + (-0.900 - 0.433i)T \)
good2 \( 1 + (1.15 - 0.785i)T + (0.730 - 1.86i)T^{2} \)
3 \( 1 + (-1.86 + 1.73i)T + (0.224 - 2.99i)T^{2} \)
5 \( 1 + (1.72 + 0.532i)T + (4.13 + 2.81i)T^{2} \)
11 \( 1 + (-0.185 - 2.48i)T + (-10.8 + 1.63i)T^{2} \)
17 \( 1 + (-7.11 - 1.07i)T + (16.2 + 5.01i)T^{2} \)
19 \( 1 + (1.55 - 2.69i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-8.21 + 1.23i)T + (21.9 - 6.77i)T^{2} \)
29 \( 1 + (5.32 - 6.67i)T + (-6.45 - 28.2i)T^{2} \)
31 \( 1 + (-4.83 - 8.37i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.590 + 1.50i)T + (-27.1 + 25.1i)T^{2} \)
41 \( 1 + (0.591 + 2.59i)T + (-36.9 + 17.7i)T^{2} \)
43 \( 1 + (-1.01 + 4.44i)T + (-38.7 - 18.6i)T^{2} \)
47 \( 1 + (-6.50 + 4.43i)T + (17.1 - 43.7i)T^{2} \)
53 \( 1 + (1.64 - 4.19i)T + (-38.8 - 36.0i)T^{2} \)
59 \( 1 + (-9.86 + 3.04i)T + (48.7 - 33.2i)T^{2} \)
61 \( 1 + (-2.14 - 5.47i)T + (-44.7 + 41.4i)T^{2} \)
67 \( 1 + (-1.07 - 1.86i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (0.825 + 1.03i)T + (-15.7 + 69.2i)T^{2} \)
73 \( 1 + (6.46 + 4.40i)T + (26.6 + 67.9i)T^{2} \)
79 \( 1 + (5.70 - 9.88i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (14.3 - 6.89i)T + (51.7 - 64.8i)T^{2} \)
89 \( 1 + (0.638 - 8.51i)T + (-88.0 - 13.2i)T^{2} \)
97 \( 1 + 14.7T + 97T^{2} \)
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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

−10.37092117298124091672176970889, −9.366507849002180069668523431385, −8.608185512507062052758160481868, −8.279413771750021691894879567959, −7.27522575959335180875423261415, −6.91184024286002761730091319225, −5.48714848123817802236597496790, −3.78936384257495246344041734005, −2.92049846025769895868427720703, −1.34368541755597349816129430778, 0.803471105647484720192474961251, 2.83084109471752752688970885066, 3.51787767316373606255172197536, 4.49102060952965724072941321548, 5.83167680899834229743100364811, 7.44860012797829854587122048465, 8.063032072580431160996202359148, 8.916057119074951200381813582550, 9.746035380579231470316656949241, 10.10587199276157693712267987077

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