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Properties

Label 2-637-91.88-c1-0-12
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: $\chi_{637} (361, \cdot )$
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

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

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