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Properties

Label 2-637-13.4-c1-0-15
Degree $2$
Conductor $637$
Sign $0.928 + 0.371i$
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  + (−0.433 − 0.249i)2-s + (−0.424 + 0.735i)3-s + (−0.875 − 1.51i)4-s − 1.04i·5-s + (0.367 − 0.212i)6-s + 1.87i·8-s + (1.13 + 1.97i)9-s + (−0.260 + 0.451i)10-s + (3.43 + 1.98i)11-s + 1.48·12-s + (−3.57 + 0.468i)13-s + (0.767 + 0.442i)15-s + (−1.28 + 2.21i)16-s + (−0.0710 − 0.123i)17-s − 1.13i·18-s + (4.77 − 2.75i)19-s + ⋯
L(s)  = 1  + (−0.306 − 0.176i)2-s + (−0.245 + 0.424i)3-s + (−0.437 − 0.757i)4-s − 0.466i·5-s + (0.150 − 0.0867i)6-s + 0.662i·8-s + (0.379 + 0.657i)9-s + (−0.0824 + 0.142i)10-s + (1.03 + 0.598i)11-s + 0.429·12-s + (−0.991 + 0.129i)13-s + (0.198 + 0.114i)15-s + (−0.320 + 0.554i)16-s + (−0.0172 − 0.0298i)17-s − 0.268i·18-s + (1.09 − 0.632i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.928 + 0.371i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.928 + 0.371i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $0.928 + 0.371i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (589, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ 0.928 + 0.371i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.08156 - 0.208332i\)
\(L(\frac12)\) \(\approx\) \(1.08156 - 0.208332i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.57 - 0.468i)T \)
good2 \( 1 + (0.433 + 0.249i)T + (1 + 1.73i)T^{2} \)
3 \( 1 + (0.424 - 0.735i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + 1.04iT - 5T^{2} \)
11 \( 1 + (-3.43 - 1.98i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (0.0710 + 0.123i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-4.77 + 2.75i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.19 + 3.80i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-4.19 + 7.27i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 2.84iT - 31T^{2} \)
37 \( 1 + (-0.730 - 0.421i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-10.4 - 6.04i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2.41 - 4.17i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 4.55iT - 47T^{2} \)
53 \( 1 + 0.279T + 53T^{2} \)
59 \( 1 + (9.33 - 5.39i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.93 - 5.07i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-4.45 - 2.57i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (3.20 - 1.84i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + 6.61iT - 73T^{2} \)
79 \( 1 - 11.9T + 79T^{2} \)
83 \( 1 + 2.87iT - 83T^{2} \)
89 \( 1 + (1.51 + 0.873i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-2.34 + 1.35i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.36105142528090435419594722295, −9.564278143754343255640967499569, −9.233839173175900122792282047441, −8.022827026997985426417870199424, −6.99244553405375949298420028250, −5.84157674600969944515844030069, −4.63464148415020974752376510276, −4.54800464525287789358321593265, −2.43266480620686340619901307089, −0.992099393083938129384946740407, 1.04493236519164816067152549530, 3.05913663715947658292060613292, 3.85214166599430489875327445638, 5.18240606633020109714091483370, 6.47982624790582097850862941076, 7.13508305135999994496736027494, 7.82051164257469918295432223378, 9.087873302934792698598721901436, 9.435745949877576154292798819781, 10.60882212055638734666416371641

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