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Properties

Label 2-637-91.74-c1-0-7
Degree $2$
Conductor $637$
Sign $0.788 - 0.615i$
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  + 0.381·2-s + (0.190 + 0.330i)3-s − 1.85·4-s + (−0.190 − 0.330i)5-s + (0.0729 + 0.126i)6-s − 1.47·8-s + (1.42 − 2.47i)9-s + (−0.0729 − 0.126i)10-s + (2.42 + 4.20i)11-s + (−0.354 − 0.613i)12-s + (−2.5 + 2.59i)13-s + (0.0729 − 0.126i)15-s + 3.14·16-s + 7.47·17-s + (0.545 − 0.944i)18-s + (−2.42 + 4.20i)19-s + ⋯
L(s)  = 1  + 0.270·2-s + (0.110 + 0.190i)3-s − 0.927·4-s + (−0.0854 − 0.147i)5-s + (0.0297 + 0.0515i)6-s − 0.520·8-s + (0.475 − 0.823i)9-s + (−0.0230 − 0.0399i)10-s + (0.731 + 1.26i)11-s + (−0.102 − 0.177i)12-s + (−0.693 + 0.720i)13-s + (0.0188 − 0.0326i)15-s + 0.786·16-s + 1.81·17-s + (0.128 − 0.222i)18-s + (−0.556 + 0.964i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.788 - 0.615i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.788 - 0.615i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $0.788 - 0.615i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (165, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ 0.788 - 0.615i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.37094 + 0.472014i\)
\(L(\frac12)\) \(\approx\) \(1.37094 + 0.472014i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.5 - 2.59i)T \)
good2 \( 1 - 0.381T + 2T^{2} \)
3 \( 1 + (-0.190 - 0.330i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (0.190 + 0.330i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-2.42 - 4.20i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 - 7.47T + 17T^{2} \)
19 \( 1 + (2.42 - 4.20i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 - 4.47T + 23T^{2} \)
29 \( 1 + (-2.04 + 3.54i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (4.35 - 7.54i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 4T + 37T^{2} \)
41 \( 1 + (2.61 - 4.53i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-3.78 - 6.54i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (1.11 + 1.93i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (4.11 - 7.13i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 - 2.23T + 59T^{2} \)
61 \( 1 + (-3 + 5.19i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.354 + 0.613i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (4.09 + 7.08i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-1 + 1.73i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (2 + 3.46i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 6.70T + 83T^{2} \)
89 \( 1 - 16.0T + 89T^{2} \)
97 \( 1 + (6.07 + 10.5i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.32400694841915228162270320184, −9.678698605158470124967534998790, −9.200037576171164035457654674440, −8.122921147715692305485499345778, −7.10702912309122237467915676185, −6.12940927769487844689010925985, −4.83890177211702428768172137540, −4.26394807553888105562467909145, −3.25070604956292780961921631338, −1.32201192923907521844642456413, 0.908292234904990279633198491028, 2.89759278150173823882256761019, 3.82910867323253268922916268207, 5.08102241373909164256810080829, 5.65800520588285851243097956677, 7.04822610718780718941328056415, 7.905777916241913606336412873079, 8.742642980836335978556712075484, 9.555707999088539856224545927887, 10.48681149232886371508973371618

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