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Properties

Label 2-637-91.54-c1-0-4
Degree $2$
Conductor $637$
Sign $-0.777 - 0.628i$
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  + (0.653 − 0.653i)2-s + (0.988 + 0.570i)3-s + 1.14i·4-s + (−3.82 + 1.02i)5-s + (1.01 − 0.272i)6-s + (2.05 + 2.05i)8-s + (−0.848 − 1.47i)9-s + (−1.82 + 3.16i)10-s + (−2.03 + 0.544i)11-s + (−0.654 + 1.13i)12-s + (−3.41 + 1.16i)13-s + (−4.36 − 1.16i)15-s + 0.391·16-s − 3.17·17-s + (−1.51 − 0.405i)18-s + (0.302 − 1.12i)19-s + ⋯
L(s)  = 1  + (0.461 − 0.461i)2-s + (0.570 + 0.329i)3-s + 0.573i·4-s + (−1.71 + 0.458i)5-s + (0.415 − 0.111i)6-s + (0.726 + 0.726i)8-s + (−0.282 − 0.490i)9-s + (−0.578 + 1.00i)10-s + (−0.613 + 0.164i)11-s + (−0.188 + 0.327i)12-s + (−0.946 + 0.324i)13-s + (−1.12 − 0.301i)15-s + 0.0979·16-s − 0.769·17-s + (−0.357 − 0.0956i)18-s + (0.0693 − 0.258i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.777 - 0.628i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.777 - 0.628i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $-0.777 - 0.628i$
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: \((2,\ 637,\ (\ :1/2),\ -0.777 - 0.628i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.267781 + 0.757543i\)
\(L(\frac12)\) \(\approx\) \(0.267781 + 0.757543i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.41 - 1.16i)T \)
good2 \( 1 + (-0.653 + 0.653i)T - 2iT^{2} \)
3 \( 1 + (-0.988 - 0.570i)T + (1.5 + 2.59i)T^{2} \)
5 \( 1 + (3.82 - 1.02i)T + (4.33 - 2.5i)T^{2} \)
11 \( 1 + (2.03 - 0.544i)T + (9.52 - 5.5i)T^{2} \)
17 \( 1 + 3.17T + 17T^{2} \)
19 \( 1 + (-0.302 + 1.12i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 - 3.76iT - 23T^{2} \)
29 \( 1 + (0.584 + 1.01i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (1.30 - 4.88i)T + (-26.8 - 15.5i)T^{2} \)
37 \( 1 + (-3.12 - 3.12i)T + 37iT^{2} \)
41 \( 1 + (1.85 - 6.93i)T + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (1.91 + 1.10i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-3.00 - 11.2i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (2.44 + 4.23i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (0.173 - 0.173i)T - 59iT^{2} \)
61 \( 1 + (-10.7 + 6.18i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.66 - 9.95i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (-1.60 - 5.98i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 + (4.75 + 1.27i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (-1.34 + 2.33i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (3.31 + 3.31i)T + 83iT^{2} \)
89 \( 1 + (-4.91 + 4.91i)T - 89iT^{2} \)
97 \( 1 + (-15.8 + 4.24i)T + (84.0 - 48.5i)T^{2} \)
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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

−11.40816633937124209424015789523, −10.20013757286813296273217706382, −9.050504891015037088595333847156, −8.181478622641926157070802567653, −7.58897293379291324842552085106, −6.76025702528368819545314366790, −4.89148549999766963185229353188, −4.15509072121317250470945799061, −3.30563189912979738801032092505, −2.59802771681015067210184653409, 0.33271769491441823553398880592, 2.35118643600213626353469683580, 3.77916360719331961916239551549, 4.72294055304510951417646432219, 5.45864937445366100910615982416, 6.91720109153311761863312280344, 7.59909412508084512880995195945, 8.222587959750663377654881529193, 9.116844839421265858661773082400, 10.41241455909888679908145440050

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