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Properties

Label 2-637-91.45-c1-0-31
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

\(L(1)\) \(\approx\) \(2.25617 - 1.02622i\)
\(L(\frac12)\) \(\approx\) \(2.25617 - 1.02622i\)
\(L(\frac{3}{2})\) 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

−10.68189100023112704147153955338, −9.388189119758517590955312106814, −8.554262585827119115645996027256, −7.993662526030603121497881931135, −7.21403453355329270230818559777, −5.63829403512273264586553270959, −4.92408684783332009419344990250, −3.30632973937514346089266885260, −3.00741191249215658065173424863, −1.35339960365311960196418434132, 1.71167120852503823628112371371, 3.11181464273617410883148894873, 4.25007247084731532655288087548, 5.09413705518551066398101455665, 6.37028613249849131260368707836, 6.92862851346305509295184166609, 7.936039031931725229526694025151, 9.282760867243276574469137374970, 9.799549257003025412367252978750, 10.30104090564642494994493494673

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