Properties

Label 2-637-91.30-c1-0-30
Degree $2$
Conductor $637$
Sign $0.923 + 0.384i$
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  + (−1.5 + 0.866i)2-s + 3-s + (0.5 − 0.866i)4-s + (1.5 + 0.866i)5-s + (−1.5 + 0.866i)6-s − 1.73i·8-s − 2·9-s − 3·10-s − 5.19i·11-s + (0.5 − 0.866i)12-s + (1 − 3.46i)13-s + (1.5 + 0.866i)15-s + (2.49 + 4.33i)16-s + (3 − 5.19i)17-s + (3 − 1.73i)18-s − 1.73i·19-s + ⋯
L(s)  = 1  + (−1.06 + 0.612i)2-s + 0.577·3-s + (0.250 − 0.433i)4-s + (0.670 + 0.387i)5-s + (−0.612 + 0.353i)6-s − 0.612i·8-s − 0.666·9-s − 0.948·10-s − 1.56i·11-s + (0.144 − 0.250i)12-s + (0.277 − 0.960i)13-s + (0.387 + 0.223i)15-s + (0.624 + 1.08i)16-s + (0.727 − 1.26i)17-s + (0.707 − 0.408i)18-s − 0.397i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.923 + 0.384i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.923 + 0.384i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $0.923 + 0.384i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (30, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ 0.923 + 0.384i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.936416 - 0.187366i\)
\(L(\frac12)\) \(\approx\) \(0.936416 - 0.187366i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1 + 3.46i)T \)
good2 \( 1 + (1.5 - 0.866i)T + (1 - 1.73i)T^{2} \)
3 \( 1 - T + 3T^{2} \)
5 \( 1 + (-1.5 - 0.866i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + 5.19iT - 11T^{2} \)
17 \( 1 + (-3 + 5.19i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + 1.73iT - 19T^{2} \)
23 \( 1 + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.5 - 2.59i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (1.5 - 0.866i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-4.5 - 2.59i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (5.5 + 9.52i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-7.5 - 4.33i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-4.5 - 7.79i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-3 - 1.73i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + 7T + 61T^{2} \)
67 \( 1 - 8.66iT - 67T^{2} \)
71 \( 1 + (-1.5 + 0.866i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (-7.5 + 4.33i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-2.5 + 4.33i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 3.46iT - 83T^{2} \)
89 \( 1 + (-6 + 3.46i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-4.5 + 2.59i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.32047630075020485040049107394, −9.325956418422110678002870577232, −8.796033487749241537021023395254, −8.042982939212086687303114056667, −7.27840674149287705930315240607, −6.11154653709719775935433643482, −5.50079942395521476887978407787, −3.51741954053204866270502564434, −2.73516923260360768465383576781, −0.70586799034918320325363066594, 1.62158487891665727214313765225, 2.23886399269075425055084671501, 3.80894551261859145822128750566, 5.15732904004356608295242730285, 6.15306792176433334455836663977, 7.53737002805107178408428701174, 8.290159621570098065336256311891, 9.153498610713197598951502863832, 9.629752016250532487112023106325, 10.30811102765878725475248472929

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