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Properties

Label 2-637-91.54-c1-0-28
Degree $2$
Conductor $637$
Sign $0.451 - 0.892i$
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  + (0.0825 − 0.0825i)2-s + (2.25 + 1.29i)3-s + 1.98i·4-s + (1.70 − 0.456i)5-s + (0.293 − 0.0785i)6-s + (0.329 + 0.329i)8-s + (1.87 + 3.25i)9-s + (0.103 − 0.178i)10-s + (1.89 − 0.506i)11-s + (−2.58 + 4.47i)12-s + (−1.85 − 3.09i)13-s + (4.43 + 1.18i)15-s − 3.91·16-s + 4.27·17-s + (0.423 + 0.113i)18-s + (−1.10 + 4.12i)19-s + ⋯
L(s)  = 1  + (0.0583 − 0.0583i)2-s + (1.29 + 0.750i)3-s + 0.993i·4-s + (0.762 − 0.204i)5-s + (0.119 − 0.0320i)6-s + (0.116 + 0.116i)8-s + (0.625 + 1.08i)9-s + (0.0325 − 0.0564i)10-s + (0.570 − 0.152i)11-s + (−0.745 + 1.29i)12-s + (−0.515 − 0.857i)13-s + (1.14 + 0.306i)15-s − 0.979·16-s + 1.03·17-s + (0.0997 + 0.0267i)18-s + (−0.253 + 0.945i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.451 - 0.892i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.451 - 0.892i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $0.451 - 0.892i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (509, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ 0.451 - 0.892i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.23095 + 1.37149i\)
\(L(\frac12)\) \(\approx\) \(2.23095 + 1.37149i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.85 + 3.09i)T \)
good2 \( 1 + (-0.0825 + 0.0825i)T - 2iT^{2} \)
3 \( 1 + (-2.25 - 1.29i)T + (1.5 + 2.59i)T^{2} \)
5 \( 1 + (-1.70 + 0.456i)T + (4.33 - 2.5i)T^{2} \)
11 \( 1 + (-1.89 + 0.506i)T + (9.52 - 5.5i)T^{2} \)
17 \( 1 - 4.27T + 17T^{2} \)
19 \( 1 + (1.10 - 4.12i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + 6.39iT - 23T^{2} \)
29 \( 1 + (3.57 + 6.19i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (1.10 - 4.13i)T + (-26.8 - 15.5i)T^{2} \)
37 \( 1 + (2.00 + 2.00i)T + 37iT^{2} \)
41 \( 1 + (2.94 - 11.0i)T + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (1.55 + 0.896i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (1.71 + 6.40i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (2.13 + 3.70i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (1.19 - 1.19i)T - 59iT^{2} \)
61 \( 1 + (2.66 - 1.54i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.00510 - 0.0190i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (-1.23 - 4.59i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 + (0.954 + 0.255i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (-2.96 + 5.14i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-9.87 - 9.87i)T + 83iT^{2} \)
89 \( 1 + (-5.68 + 5.68i)T - 89iT^{2} \)
97 \( 1 + (-14.2 + 3.82i)T + (84.0 - 48.5i)T^{2} \)
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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

−10.31131761648005009616089966185, −9.816735423036536179888623357124, −8.950259263573454675496858351487, −8.221923421379620818172243429702, −7.62898129282551040582263250555, −6.24195546844585938148086059091, −4.97211062679823339962157660913, −3.85419428445171344561847503732, −3.15170777435124031724402291150, −2.08818909392167924872484675345, 1.51649738226384055302176410508, 2.20543959615622610179823527320, 3.54534801911346458064325717994, 5.00154729485815012982415627950, 6.06460278614675930414860151346, 6.97360186994615038554277460161, 7.61829763177641879874647041644, 9.060248130596778546079295961155, 9.302177579346806800295286533798, 10.14057748101005552596881500690

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