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Properties

Label 2-637-91.81-c1-0-9
Degree $2$
Conductor $637$
Sign $0.444 - 0.895i$
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  + (−0.929 + 1.60i)2-s − 2.29·3-s + (−0.726 − 1.25i)4-s + (−0.0986 − 0.170i)5-s + (2.13 − 3.69i)6-s − 1.01·8-s + 2.26·9-s + 0.366·10-s − 4.18·11-s + (1.66 + 2.88i)12-s + (2.72 − 2.36i)13-s + (0.226 + 0.392i)15-s + (2.39 − 4.15i)16-s + (0.420 + 0.728i)17-s + (−2.10 + 3.64i)18-s − 1.35·19-s + ⋯
L(s)  = 1  + (−0.656 + 1.13i)2-s − 1.32·3-s + (−0.363 − 0.629i)4-s + (−0.0441 − 0.0764i)5-s + (0.870 − 1.50i)6-s − 0.359·8-s + 0.754·9-s + 0.115·10-s − 1.26·11-s + (0.481 + 0.833i)12-s + (0.755 − 0.655i)13-s + (0.0584 + 0.101i)15-s + (0.599 − 1.03i)16-s + (0.102 + 0.176i)17-s + (−0.495 + 0.858i)18-s − 0.310·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.444 - 0.895i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.444 - 0.895i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $0.444 - 0.895i$
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: \((2,\ 637,\ (\ :1/2),\ 0.444 - 0.895i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.417700 + 0.259004i\)
\(L(\frac12)\) \(\approx\) \(0.417700 + 0.259004i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.72 + 2.36i)T \)
good2 \( 1 + (0.929 - 1.60i)T + (-1 - 1.73i)T^{2} \)
3 \( 1 + 2.29T + 3T^{2} \)
5 \( 1 + (0.0986 + 0.170i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + 4.18T + 11T^{2} \)
17 \( 1 + (-0.420 - 0.728i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + 1.35T + 19T^{2} \)
23 \( 1 + (-2.05 + 3.56i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-4.11 - 7.13i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (0.640 - 1.10i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (1.52 - 2.63i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-2.69 - 4.67i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (2.66 - 4.61i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (5.83 + 10.1i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (2.32 - 4.02i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-3.02 - 5.24i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 - 11.3T + 61T^{2} \)
67 \( 1 - 13.3T + 67T^{2} \)
71 \( 1 + (-2.98 + 5.17i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-1.94 + 3.36i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-5.36 - 9.29i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 3.07T + 83T^{2} \)
89 \( 1 + (5.99 - 10.3i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-9.73 + 16.8i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.60261076749472649313254067339, −10.00797752538361184289109113886, −8.530685743955016185515095972544, −8.228235512321743715713146976875, −6.96505278064917067101868392447, −6.38695117895641645578782567915, −5.48884597918814251161430821964, −4.86016571672311424839571511200, −3.00988501971477982357479456417, −0.63809180421809327844486705563, 0.75927074381916586287914175780, 2.22913426550544919762513440938, 3.55667181735912734254493668608, 4.97395509334995455314010015158, 5.83173005514753618630336178014, 6.73176710573824770841867295558, 7.982223404403285015083321281312, 8.979714041178771207655058581326, 9.931629227384314432877592037644, 10.59868014875479028472497290776

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