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Properties

Label 2-637-91.30-c1-0-5
Degree $2$
Conductor $637$
Sign $-0.803 - 0.595i$
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  + (0.104 − 0.0601i)2-s − 0.582·3-s + (−0.992 + 1.71i)4-s + (1.46 + 0.844i)5-s + (−0.0606 + 0.0350i)6-s + 0.479i·8-s − 2.66·9-s + 0.203·10-s + 0.364i·11-s + (0.578 − 1.00i)12-s + (1.80 + 3.12i)13-s + (−0.851 − 0.491i)15-s + (−1.95 − 3.38i)16-s + (−1.59 + 2.75i)17-s + (−0.277 + 0.160i)18-s + 1.44i·19-s + ⋯
L(s)  = 1  + (0.0737 − 0.0425i)2-s − 0.336·3-s + (−0.496 + 0.859i)4-s + (0.653 + 0.377i)5-s + (−0.0247 + 0.0143i)6-s + 0.169i·8-s − 0.886·9-s + 0.0642·10-s + 0.109i·11-s + (0.166 − 0.289i)12-s + (0.499 + 0.866i)13-s + (−0.219 − 0.126i)15-s + (−0.489 − 0.847i)16-s + (−0.386 + 0.669i)17-s + (−0.0653 + 0.0377i)18-s + 0.331i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.803 - 0.595i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.803 - 0.595i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $-0.803 - 0.595i$
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: \((2,\ 637,\ (\ :1/2),\ -0.803 - 0.595i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.255722 + 0.774238i\)
\(L(\frac12)\) \(\approx\) \(0.255722 + 0.774238i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.80 - 3.12i)T \)
good2 \( 1 + (-0.104 + 0.0601i)T + (1 - 1.73i)T^{2} \)
3 \( 1 + 0.582T + 3T^{2} \)
5 \( 1 + (-1.46 - 0.844i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 - 0.364iT - 11T^{2} \)
17 \( 1 + (1.59 - 2.75i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 - 1.44iT - 19T^{2} \)
23 \( 1 + (2.54 + 4.40i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (4.09 - 7.09i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (4.06 - 2.34i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (5.46 - 3.15i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (5.04 + 2.91i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.386 + 0.669i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-11.0 - 6.39i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (0.685 + 1.18i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-8.10 - 4.68i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + 9.02T + 61T^{2} \)
67 \( 1 - 13.4iT - 67T^{2} \)
71 \( 1 + (6.13 - 3.54i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (-1.87 + 1.08i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-3.44 + 5.96i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 0.567iT - 83T^{2} \)
89 \( 1 + (-0.986 + 0.569i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-6.86 + 3.96i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.92042356572745127521299243674, −10.17488738792584712503252112776, −8.837528465479733513914382839333, −8.679317304578107361711609026619, −7.33664763646722801486946316250, −6.40212404750980619377133905724, −5.53744488192908569789708593536, −4.35958716397527796243396616314, −3.33491408374266540928933268194, −2.07597237543405583232259579481, 0.43665822009123183739497436232, 2.01374460532462980798950292803, 3.63311802568387077990621226236, 5.04634122538000014105213944125, 5.62728434136593061598897418578, 6.22595750523641561649868270682, 7.61559716099771872530647187908, 8.782074820827153502575510390330, 9.343541464337046782152514790534, 10.20086394438805514216347831171

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