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Properties

Label 2-637-13.9-c1-0-4
Degree $2$
Conductor $637$
Sign $0.859 - 0.511i$
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  + (1.20 − 2.09i)2-s + (−0.707 + 1.22i)3-s + (−1.91 − 3.31i)4-s − 3.82·5-s + (1.70 + 2.95i)6-s − 4.41·8-s + (0.500 + 0.866i)9-s + (−4.62 + 8.00i)10-s + (−1.70 + 2.95i)11-s + 5.41·12-s + (3.5 + 0.866i)13-s + (2.70 − 4.68i)15-s + (−1.49 + 2.59i)16-s + (−0.0857 − 0.148i)17-s + 2.41·18-s + (3 + 5.19i)19-s + ⋯
L(s)  = 1  + (0.853 − 1.47i)2-s + (−0.408 + 0.707i)3-s + (−0.957 − 1.65i)4-s − 1.71·5-s + (0.696 + 1.20i)6-s − 1.56·8-s + (0.166 + 0.288i)9-s + (−1.46 + 2.53i)10-s + (−0.514 + 0.891i)11-s + 1.56·12-s + (0.970 + 0.240i)13-s + (0.698 − 1.21i)15-s + (−0.374 + 0.649i)16-s + (−0.0208 − 0.0360i)17-s + 0.569·18-s + (0.688 + 1.19i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.859 - 0.511i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.859 - 0.511i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $0.859 - 0.511i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (295, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ 0.859 - 0.511i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.889822 + 0.244552i\)
\(L(\frac12)\) \(\approx\) \(0.889822 + 0.244552i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.5 - 0.866i)T \)
good2 \( 1 + (-1.20 + 2.09i)T + (-1 - 1.73i)T^{2} \)
3 \( 1 + (0.707 - 1.22i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + 3.82T + 5T^{2} \)
11 \( 1 + (1.70 - 2.95i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (0.0857 + 0.148i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3 - 5.19i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.707 - 1.22i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (4.91 - 8.51i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 5.41T + 31T^{2} \)
37 \( 1 + (3.74 - 6.48i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-2.91 + 5.04i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.292 + 0.507i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 7.65T + 47T^{2} \)
53 \( 1 + 3T + 53T^{2} \)
59 \( 1 + (0.878 + 1.52i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4.91 - 8.51i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.12 + 3.67i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-0.171 - 0.297i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + 0.656T + 73T^{2} \)
79 \( 1 - 10.2T + 79T^{2} \)
83 \( 1 - 13.0T + 83T^{2} \)
89 \( 1 + (3.65 - 6.33i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (2.58 + 4.47i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.90237668753824468454400666970, −10.28956228087451269779582249135, −9.336020173556148288730396704848, −8.068143082308303798787993203638, −7.19874146076815876915081218209, −5.45316014604351275591986321742, −4.76694037878526961647975600959, −3.84223910874906252569008324359, −3.43001557290617064365149736583, −1.64863558962648945417496144642, 0.41931689561002645733451262767, 3.38808143717308652013818923618, 4.04288671912314537076766637627, 5.20864217871328387909301825506, 6.13005199983438260811236689557, 6.91609129150245368636605409471, 7.71492666965510169771977848986, 8.114874110951909686784647368284, 9.137354885807611651043164392560, 11.04625814875401754414137316837

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