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Properties

Label 2-637-91.24-c1-0-20
Degree $2$
Conductor $637$
Sign $0.747 - 0.664i$
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  + (−0.813 + 0.218i)2-s + 1.43i·3-s + (−1.11 + 0.645i)4-s + (1.38 + 0.370i)5-s + (−0.312 − 1.16i)6-s + (1.95 − 1.95i)8-s + 0.945·9-s − 1.20·10-s + (3.43 − 3.43i)11-s + (−0.924 − 1.60i)12-s + (1.04 − 3.44i)13-s + (−0.531 + 1.98i)15-s + (0.122 − 0.212i)16-s + (1.49 + 2.58i)17-s + (−0.769 + 0.206i)18-s + (4.44 − 4.44i)19-s + ⋯
L(s)  = 1  + (−0.575 + 0.154i)2-s + 0.827i·3-s + (−0.558 + 0.322i)4-s + (0.618 + 0.165i)5-s + (−0.127 − 0.476i)6-s + (0.692 − 0.692i)8-s + 0.315·9-s − 0.381·10-s + (1.03 − 1.03i)11-s + (−0.266 − 0.462i)12-s + (0.291 − 0.956i)13-s + (−0.137 + 0.512i)15-s + (0.0306 − 0.0531i)16-s + (0.361 + 0.626i)17-s + (−0.181 + 0.0486i)18-s + (1.01 − 1.01i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.747 - 0.664i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.747 - 0.664i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $0.747 - 0.664i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (570, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ 0.747 - 0.664i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.14083 + 0.433985i\)
\(L(\frac12)\) \(\approx\) \(1.14083 + 0.433985i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.813 - 0.218i)T + (1.73 - i)T^{2} \)
3 \( 1 - 1.43iT - 3T^{2} \)
5 \( 1 + (-1.38 - 0.370i)T + (4.33 + 2.5i)T^{2} \)
11 \( 1 + (-3.43 + 3.43i)T - 11iT^{2} \)
17 \( 1 + (-1.49 - 2.58i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-4.44 + 4.44i)T - 19iT^{2} \)
23 \( 1 + (-1.02 - 0.590i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (2.77 + 4.81i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-1.75 - 6.54i)T + (-26.8 + 15.5i)T^{2} \)
37 \( 1 + (1.44 + 5.40i)T + (-32.0 + 18.5i)T^{2} \)
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 \( 1 + (1.51 - 5.63i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (2.89 - 5.01i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-2.88 + 10.7i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 - 9.18iT - 61T^{2} \)
67 \( 1 + (1.38 + 1.38i)T + 67iT^{2} \)
71 \( 1 + (3.19 - 0.855i)T + (61.4 - 35.5i)T^{2} \)
73 \( 1 + (-0.482 + 0.129i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (-3.47 - 6.02i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-3.22 + 3.22i)T - 83iT^{2} \)
89 \( 1 + (-0.237 + 0.0636i)T + (77.0 - 44.5i)T^{2} \)
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

−10.43896554786928029392780016857, −9.643642938295551583629339657051, −9.157735710196887846226544994727, −8.284968793495691279545275473337, −7.31290709628784405752735027429, −6.14529626321652915407016961905, −5.16310422829106225294650915923, −4.01092803804708337390281783924, −3.22873832238053995553807450143, −1.08673602042355269346164385694, 1.29675642036335812524023713524, 1.88984320761466605230516939454, 3.94747181500667724491954061524, 4.99831084815299381716780006603, 6.10120648065163895160268288115, 7.07161259777384025718653031312, 7.81364192511976583186204471187, 8.969451540838309314085717442923, 9.672536899813159787345560800942, 10.02316892501617450364613218495

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