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Properties

Label 2-637-91.19-c1-0-29
Degree $2$
Conductor $637$
Sign $-0.692 + 0.721i$
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  + (−2.28 − 0.612i)2-s − 1.20i·3-s + (3.12 + 1.80i)4-s + (3.08 − 0.825i)5-s + (−0.736 + 2.74i)6-s + (−2.68 − 2.68i)8-s + 1.55·9-s − 7.55·10-s + (−3.00 − 3.00i)11-s + (2.16 − 3.75i)12-s + (−3.48 + 0.920i)13-s + (−0.992 − 3.70i)15-s + (0.897 + 1.55i)16-s + (−0.721 + 1.25i)17-s + (−3.55 − 0.952i)18-s + (−1.77 − 1.77i)19-s + ⋯
L(s)  = 1  + (−1.61 − 0.433i)2-s − 0.694i·3-s + (1.56 + 0.901i)4-s + (1.37 − 0.369i)5-s + (−0.300 + 1.12i)6-s + (−0.950 − 0.950i)8-s + 0.518·9-s − 2.38·10-s + (−0.907 − 0.907i)11-s + (0.625 − 1.08i)12-s + (−0.966 + 0.255i)13-s + (−0.256 − 0.956i)15-s + (0.224 + 0.388i)16-s + (−0.175 + 0.303i)17-s + (−0.838 − 0.224i)18-s + (−0.407 − 0.407i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.692 + 0.721i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.692 + 0.721i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $-0.692 + 0.721i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ -0.692 + 0.721i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.296099 - 0.694041i\)
\(L(\frac12)\) \(\approx\) \(0.296099 - 0.694041i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.48 - 0.920i)T \)
good2 \( 1 + (2.28 + 0.612i)T + (1.73 + i)T^{2} \)
3 \( 1 + 1.20iT - 3T^{2} \)
5 \( 1 + (-3.08 + 0.825i)T + (4.33 - 2.5i)T^{2} \)
11 \( 1 + (3.00 + 3.00i)T + 11iT^{2} \)
17 \( 1 + (0.721 - 1.25i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.77 + 1.77i)T + 19iT^{2} \)
23 \( 1 + (-4.52 + 2.61i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.34 + 2.32i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-1.37 + 5.14i)T + (-26.8 - 15.5i)T^{2} \)
37 \( 1 + (-0.160 + 0.599i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + (-5.04 + 1.35i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (5.46 - 3.15i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (1.71 + 6.39i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (-3.79 - 6.56i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (0.525 + 1.96i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + 5.24iT - 61T^{2} \)
67 \( 1 + (-6.19 + 6.19i)T - 67iT^{2} \)
71 \( 1 + (-8.31 - 2.22i)T + (61.4 + 35.5i)T^{2} \)
73 \( 1 + (15.2 + 4.09i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (1.00 - 1.74i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (5.11 + 5.11i)T + 83iT^{2} \)
89 \( 1 + (6.76 + 1.81i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (1.62 - 6.06i)T + (-84.0 - 48.5i)T^{2} \)
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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

−10.10254951025081065307683628727, −9.457181267665016760557096530026, −8.652925898193677508818859354990, −7.84903469820069189604345014224, −6.95591412779922804177521509156, −6.08580717425563384060858886692, −4.85226556948118931830931713891, −2.63658341024523120641968472848, −1.96615619161192180033764062257, −0.69139563811071842648600993107, 1.63278512981642103581588601402, 2.70749939738390812551691830180, 4.72081215409685768668789136583, 5.62127891689057400409761735454, 6.89053991496401099935843282324, 7.30399822342786699460469715902, 8.508882541655533427875025854520, 9.471217133661742993526491405508, 9.956097630299220851950842785885, 10.29208076727933527577987831109

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