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Properties

Label 2-637-91.30-c1-0-42
Degree $2$
Conductor $637$
Sign $-0.915 + 0.403i$
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  + (2.34 − 1.35i)2-s − 0.345·3-s + (2.65 − 4.59i)4-s + (−2.82 − 1.62i)5-s + (−0.809 + 0.467i)6-s − 8.94i·8-s − 2.88·9-s − 8.80·10-s + 1.84i·11-s + (−0.918 + 1.59i)12-s + (3.60 − 0.0186i)13-s + (0.976 + 0.563i)15-s + (−6.77 − 11.7i)16-s + (1.07 − 1.86i)17-s + (−6.74 + 3.89i)18-s − 2.40i·19-s + ⋯
L(s)  = 1  + (1.65 − 0.955i)2-s − 0.199·3-s + (1.32 − 2.29i)4-s + (−1.26 − 0.728i)5-s + (−0.330 + 0.190i)6-s − 3.16i·8-s − 0.960·9-s − 2.78·10-s + 0.556i·11-s + (−0.265 + 0.459i)12-s + (0.999 − 0.00517i)13-s + (0.252 + 0.145i)15-s + (−1.69 − 2.93i)16-s + (0.261 − 0.452i)17-s + (−1.58 + 0.917i)18-s − 0.550i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.915 + 0.403i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.915 + 0.403i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $-0.915 + 0.403i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (30, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ -0.915 + 0.403i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.542002 - 2.57546i\)
\(L(\frac12)\) \(\approx\) \(0.542002 - 2.57546i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.60 + 0.0186i)T \)
good2 \( 1 + (-2.34 + 1.35i)T + (1 - 1.73i)T^{2} \)
3 \( 1 + 0.345T + 3T^{2} \)
5 \( 1 + (2.82 + 1.62i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 - 1.84iT - 11T^{2} \)
17 \( 1 + (-1.07 + 1.86i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + 2.40iT - 19T^{2} \)
23 \( 1 + (-0.906 - 1.56i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.36 + 2.36i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-1.50 + 0.871i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-5.14 + 2.96i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-3.65 - 2.11i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (4.34 + 7.51i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-5.09 - 2.93i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-4.65 - 8.05i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (9.31 + 5.37i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 - 10.1T + 61T^{2} \)
67 \( 1 + 0.826iT - 67T^{2} \)
71 \( 1 + (2.03 - 1.17i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (2.76 - 1.59i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (0.400 - 0.694i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 9.97iT - 83T^{2} \)
89 \( 1 + (13.0 - 7.55i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-7.99 + 4.61i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.84559872365773469317627891546, −9.582062024583436520509753339915, −8.509962331718766657789011602369, −7.32693458308039854584364613878, −6.15430902618812318134538176497, −5.28416773180780346763309920308, −4.43498071825768411912281081394, −3.69153147795801853344266961871, −2.61548574411380848360363859281, −0.891702457905492016387122306263, 2.93871175384895352020803147107, 3.54418810326667881017476104109, 4.43269772768536708289034115313, 5.67189418910440897208967963241, 6.27554312072061100261203327664, 7.14710790200104237161229239588, 8.121751262264026581669465930847, 8.518384134106851110950343845357, 10.66117030754739795310000141961, 11.37934985576442217797172726616

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