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Properties

Label 2-637-91.30-c1-0-25
Degree $2$
Conductor $637$
Sign $0.0162 + 0.999i$
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  + (0.433 − 0.249i)2-s − 0.849·3-s + (−0.875 + 1.51i)4-s + (−0.902 − 0.521i)5-s + (−0.367 + 0.212i)6-s + 1.87i·8-s − 2.27·9-s − 0.521·10-s − 3.96i·11-s + (0.743 − 1.28i)12-s + (3.57 − 0.468i)13-s + (0.767 + 0.442i)15-s + (−1.28 − 2.21i)16-s + (0.0710 − 0.123i)17-s + (−0.986 + 0.569i)18-s − 5.50i·19-s + ⋯
L(s)  = 1  + (0.306 − 0.176i)2-s − 0.490·3-s + (−0.437 + 0.757i)4-s + (−0.403 − 0.233i)5-s + (−0.150 + 0.0867i)6-s + 0.662i·8-s − 0.759·9-s − 0.164·10-s − 1.19i·11-s + (0.214 − 0.371i)12-s + (0.991 − 0.129i)13-s + (0.198 + 0.114i)15-s + (−0.320 − 0.554i)16-s + (0.0172 − 0.0298i)17-s + (−0.232 + 0.134i)18-s − 1.26i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0162 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0162 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $0.0162 + 0.999i$
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: \((2,\ 637,\ (\ :1/2),\ 0.0162 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.584398 - 0.574968i\)
\(L(\frac12)\) \(\approx\) \(0.584398 - 0.574968i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.57 + 0.468i)T \)
good2 \( 1 + (-0.433 + 0.249i)T + (1 - 1.73i)T^{2} \)
3 \( 1 + 0.849T + 3T^{2} \)
5 \( 1 + (0.902 + 0.521i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + 3.96iT - 11T^{2} \)
17 \( 1 + (-0.0710 + 0.123i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + 5.50iT - 19T^{2} \)
23 \( 1 + (-2.19 - 3.80i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-4.19 + 7.27i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-2.46 + 1.42i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.730 - 0.421i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (10.4 + 6.04i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2.41 - 4.17i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (3.94 + 2.27i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-0.139 - 0.242i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (9.33 + 5.39i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 - 5.86T + 61T^{2} \)
67 \( 1 + 5.14iT - 67T^{2} \)
71 \( 1 + (3.20 - 1.84i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (-5.72 + 3.30i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (5.96 - 10.3i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 2.87iT - 83T^{2} \)
89 \( 1 + (1.51 - 0.873i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (2.34 - 1.35i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.70922864763336130034902181486, −9.304547247017943785568556974572, −8.426502203607531808755050103778, −8.090170653448728796770989651804, −6.64670409792323105721770068302, −5.69749976438035529320002367924, −4.77313722343878702010556312202, −3.67558977727243828473165097363, −2.81248973037168621416710412182, −0.46236738645008438634653233665, 1.45972576202296593720686490771, 3.33796888647646719149177098271, 4.48340031549164163761252439864, 5.32064876733944129084446164800, 6.24271852812818350052301753495, 6.95774418976259356707005858532, 8.250931023054810157265699340632, 9.051517152578113487177950166218, 10.16937934979264239757993256618, 10.66233658796319494928380573739

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