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Properties

Label 2-637-91.45-c1-0-8
Degree $2$
Conductor $637$
Sign $-0.657 - 0.753i$
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  + (0.595 − 0.595i)2-s + (−1.24 + 0.716i)3-s + 1.29i·4-s + (−0.370 + 1.38i)5-s + (−0.312 + 1.16i)6-s + (1.95 + 1.95i)8-s + (−0.472 + 0.819i)9-s + (0.603 + 1.04i)10-s + (1.25 − 4.68i)11-s + (−0.924 − 1.60i)12-s + (1.04 + 3.44i)13-s + (−0.531 − 1.98i)15-s − 0.245·16-s − 2.98·17-s + (0.206 + 0.769i)18-s + (−6.06 + 1.62i)19-s + ⋯
L(s)  = 1  + (0.421 − 0.421i)2-s + (−0.716 + 0.413i)3-s + 0.645i·4-s + (−0.165 + 0.618i)5-s + (−0.127 + 0.476i)6-s + (0.692 + 0.692i)8-s + (−0.157 + 0.273i)9-s + (0.190 + 0.330i)10-s + (0.378 − 1.41i)11-s + (−0.266 − 0.462i)12-s + (0.291 + 0.956i)13-s + (−0.137 − 0.512i)15-s − 0.0613·16-s − 0.723·17-s + (0.0486 + 0.181i)18-s + (−1.39 + 0.372i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.657 - 0.753i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.657 - 0.753i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $-0.657 - 0.753i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (227, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ -0.657 - 0.753i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.412556 + 0.907015i\)
\(L(\frac12)\) \(\approx\) \(0.412556 + 0.907015i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.04 - 3.44i)T \)
good2 \( 1 + (-0.595 + 0.595i)T - 2iT^{2} \)
3 \( 1 + (1.24 - 0.716i)T + (1.5 - 2.59i)T^{2} \)
5 \( 1 + (0.370 - 1.38i)T + (-4.33 - 2.5i)T^{2} \)
11 \( 1 + (-1.25 + 4.68i)T + (-9.52 - 5.5i)T^{2} \)
17 \( 1 + 2.98T + 17T^{2} \)
19 \( 1 + (6.06 - 1.62i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 - 1.18iT - 23T^{2} \)
29 \( 1 + (2.77 - 4.81i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (6.54 - 1.75i)T + (26.8 - 15.5i)T^{2} \)
37 \( 1 + (3.95 + 3.95i)T + 37iT^{2} \)
41 \( 1 + (-4.71 + 1.26i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (2.90 - 1.67i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-5.63 - 1.51i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (2.89 - 5.01i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-7.87 + 7.87i)T - 59iT^{2} \)
61 \( 1 + (-7.95 - 4.59i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.89 - 0.508i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + (3.19 + 0.855i)T + (61.4 + 35.5i)T^{2} \)
73 \( 1 + (0.129 + 0.482i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (-3.47 - 6.02i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-3.22 - 3.22i)T + 83iT^{2} \)
89 \( 1 + (0.173 - 0.173i)T - 89iT^{2} \)
97 \( 1 + (-2.43 + 9.08i)T + (-84.0 - 48.5i)T^{2} \)
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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

−11.05070197090713103631670043297, −10.64201920500501018472252576000, −9.010914578523187003673873561740, −8.455783418509819314097178414872, −7.22601105152752925235589988009, −6.33360181095639665873793026675, −5.32150165372368625562036758910, −4.17816961763980571353233988189, −3.47200342333253192063484430877, −2.15768795119306953245653621549, 0.50285636346367147671692298762, 1.96541326962520963041720453330, 4.05791160464109622582602225446, 4.86141699464071152201900792753, 5.74976953018572846906135556802, 6.57893834976668700926335597982, 7.21741287606257802832218489495, 8.520761006832963555988729934891, 9.410547769226959450958743634129, 10.38106161105717844499862059456

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