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Properties

Label 2-637-91.81-c1-0-38
Degree $2$
Conductor $637$
Sign $-0.945 - 0.324i$
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  + (0.952 − 1.65i)2-s − 0.428·3-s + (−0.815 − 1.41i)4-s + (−0.736 − 1.27i)5-s + (−0.408 + 0.707i)6-s + 0.702·8-s − 2.81·9-s − 2.80·10-s − 4.39·11-s + (0.349 + 0.605i)12-s + (−2.69 − 2.39i)13-s + (0.315 + 0.546i)15-s + (2.30 − 3.98i)16-s + (−0.601 − 1.04i)17-s + (−2.68 + 4.64i)18-s − 3.24·19-s + ⋯
L(s)  = 1  + (0.673 − 1.16i)2-s − 0.247·3-s + (−0.407 − 0.706i)4-s + (−0.329 − 0.570i)5-s + (−0.166 + 0.288i)6-s + 0.248·8-s − 0.938·9-s − 0.887·10-s − 1.32·11-s + (0.100 + 0.174i)12-s + (−0.748 − 0.663i)13-s + (0.0814 + 0.141i)15-s + (0.575 − 0.996i)16-s + (−0.145 − 0.252i)17-s + (−0.632 + 1.09i)18-s − 0.743·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.945 - 0.324i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.945 - 0.324i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $-0.945 - 0.324i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (263, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ -0.945 - 0.324i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.179262 + 1.07472i\)
\(L(\frac12)\) \(\approx\) \(0.179262 + 1.07472i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.69 + 2.39i)T \)
good2 \( 1 + (-0.952 + 1.65i)T + (-1 - 1.73i)T^{2} \)
3 \( 1 + 0.428T + 3T^{2} \)
5 \( 1 + (0.736 + 1.27i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + 4.39T + 11T^{2} \)
17 \( 1 + (0.601 + 1.04i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + 3.24T + 19T^{2} \)
23 \( 1 + (-2.21 + 3.84i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.0837 + 0.145i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-2.62 + 4.54i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (3.52 - 6.10i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-2.58 - 4.47i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.0113 - 0.0197i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-5.84 - 10.1i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-0.0708 + 0.122i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (2.67 + 4.62i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + 11.5T + 61T^{2} \)
67 \( 1 - 4.13T + 67T^{2} \)
71 \( 1 + (-4.98 + 8.63i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-7.62 + 13.1i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (0.387 + 0.670i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 16.0T + 83T^{2} \)
89 \( 1 + (-3.27 + 5.67i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-1.74 + 3.02i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.63062547657947206658950306623, −9.499994751992046781428825314314, −8.292314692253995906070624434559, −7.70352966555684094511418935660, −6.18400074663107096650075953453, −4.99513731169067331876049974783, −4.61966729873464700762909776761, −3.07970753658067069940177130467, −2.41416561476387184589168298948, −0.45856398148055129751420493950, 2.43916455650666546605227780824, 3.76172502225206942358176853142, 5.03095422333735985799967314034, 5.54398250850027135030393305099, 6.64509495059274123615325153671, 7.28746028163385941838903863413, 8.099480044760222247307224228332, 9.058489944929014379878843622457, 10.51472972612008578430295019552, 10.88845764542980333872520648132

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