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Properties

Label 2-637-91.59-c1-0-32
Degree $2$
Conductor $637$
Sign $0.259 - 0.965i$
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  + (1.81 + 1.81i)2-s + (2.33 − 1.34i)3-s + 4.59i·4-s + (0.384 + 0.103i)5-s + (6.68 + 1.79i)6-s + (−4.70 + 4.70i)8-s + (2.13 − 3.69i)9-s + (0.511 + 0.885i)10-s + (−3.50 − 0.939i)11-s + (6.19 + 10.7i)12-s + (3.44 − 1.06i)13-s + (1.03 − 0.277i)15-s − 7.90·16-s − 4.09·17-s + (10.5 − 2.83i)18-s + (−0.208 − 0.777i)19-s + ⋯
L(s)  = 1  + (1.28 + 1.28i)2-s + (1.34 − 0.778i)3-s + 2.29i·4-s + (0.172 + 0.0461i)5-s + (2.72 + 0.731i)6-s + (−1.66 + 1.66i)8-s + (0.711 − 1.23i)9-s + (0.161 + 0.280i)10-s + (−1.05 − 0.283i)11-s + (1.78 + 3.09i)12-s + (0.955 − 0.294i)13-s + (0.267 − 0.0717i)15-s − 1.97·16-s − 0.993·17-s + (2.49 − 0.668i)18-s + (−0.0478 − 0.178i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.259 - 0.965i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.259 - 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $0.259 - 0.965i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (423, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ 0.259 - 0.965i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.31613 + 2.54323i\)
\(L(\frac12)\) \(\approx\) \(3.31613 + 2.54323i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.44 + 1.06i)T \)
good2 \( 1 + (-1.81 - 1.81i)T + 2iT^{2} \)
3 \( 1 + (-2.33 + 1.34i)T + (1.5 - 2.59i)T^{2} \)
5 \( 1 + (-0.384 - 0.103i)T + (4.33 + 2.5i)T^{2} \)
11 \( 1 + (3.50 + 0.939i)T + (9.52 + 5.5i)T^{2} \)
17 \( 1 + 4.09T + 17T^{2} \)
19 \( 1 + (0.208 + 0.777i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 - 5.09iT - 23T^{2} \)
29 \( 1 + (-1.00 + 1.74i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (1.62 + 6.06i)T + (-26.8 + 15.5i)T^{2} \)
37 \( 1 + (-1.26 + 1.26i)T - 37iT^{2} \)
41 \( 1 + (-0.578 - 2.15i)T + (-35.5 + 20.5i)T^{2} \)
43 \( 1 + (2.65 - 1.53i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-2.19 + 8.19i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (4.54 - 7.87i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-5.25 - 5.25i)T + 59iT^{2} \)
61 \( 1 + (2.40 + 1.38i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.30 + 4.85i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + (0.582 - 2.17i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 + (-4.78 + 1.28i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (-1.80 - 3.13i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-5.36 + 5.36i)T - 83iT^{2} \)
89 \( 1 + (-8.44 - 8.44i)T + 89iT^{2} \)
97 \( 1 + (7.71 + 2.06i)T + (84.0 + 48.5i)T^{2} \)
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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

−10.96340358541515272339660597473, −9.453429769570230234842552467402, −8.418829332032834294214811568786, −7.968990384633047151251971674431, −7.24110413309657270412541568121, −6.29438476625127345344549865687, −5.50226771329995348111643664469, −4.18412152919424446214248669246, −3.26499389610623280905516094175, −2.27665286795862051792065361959, 1.88953583042435451493249601886, 2.74933686647479726231685512236, 3.64537918731789456518117037835, 4.43419495783935468105548141454, 5.27320176352994121875413294234, 6.52796107807166343396623296517, 8.091383929739466176594723679249, 8.973383141845427840128158937259, 9.780910355502656405736291650069, 10.57620840370255955188848263141

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