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Properties

Label 2-637-91.25-c1-0-24
Degree $2$
Conductor $637$
Sign $0.988 - 0.154i$
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  + (−0.287 + 0.166i)2-s + (0.729 − 1.26i)3-s + (−0.944 + 1.63i)4-s + (1.25 − 0.722i)5-s + 0.485i·6-s − 1.29i·8-s + (0.434 + 0.752i)9-s + (−0.240 + 0.416i)10-s + (5.15 + 2.97i)11-s + (1.37 + 2.38i)12-s + (−1.88 − 3.07i)13-s − 2.11i·15-s + (−1.67 − 2.90i)16-s + (−2.16 + 3.74i)17-s + (−0.250 − 0.144i)18-s + (1.69 − 0.978i)19-s + ⋯
L(s)  = 1  + (−0.203 + 0.117i)2-s + (0.421 − 0.729i)3-s + (−0.472 + 0.818i)4-s + (0.559 − 0.323i)5-s + 0.198i·6-s − 0.457i·8-s + (0.144 + 0.250i)9-s + (−0.0759 + 0.131i)10-s + (1.55 + 0.897i)11-s + (0.398 + 0.689i)12-s + (−0.524 − 0.851i)13-s − 0.544i·15-s + (−0.418 − 0.725i)16-s + (−0.524 + 0.909i)17-s + (−0.0589 − 0.0340i)18-s + (0.388 − 0.224i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.988 - 0.154i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.988 - 0.154i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $0.988 - 0.154i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (116, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ 0.988 - 0.154i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.63297 + 0.126615i\)
\(L(\frac12)\) \(\approx\) \(1.63297 + 0.126615i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.88 + 3.07i)T \)
good2 \( 1 + (0.287 - 0.166i)T + (1 - 1.73i)T^{2} \)
3 \( 1 + (-0.729 + 1.26i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (-1.25 + 0.722i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (-5.15 - 2.97i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (2.16 - 3.74i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.69 + 0.978i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.270 + 0.467i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 7.15T + 29T^{2} \)
31 \( 1 + (-5.28 - 3.05i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-6.95 + 4.01i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + 7.55iT - 41T^{2} \)
43 \( 1 + 4.24T + 43T^{2} \)
47 \( 1 + (5.42 - 3.13i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-1.38 + 2.40i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (0.737 + 0.425i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-3.38 - 5.87i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.854 + 0.493i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 3.76iT - 71T^{2} \)
73 \( 1 + (7.91 + 4.56i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-0.0655 - 0.113i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 2.66iT - 83T^{2} \)
89 \( 1 + (8.41 - 4.85i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 6.58iT - 97T^{2} \)
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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

−10.33264564023509171877814817718, −9.543387066643874546816154011333, −8.762595191393742245027918197038, −8.016719086566563797016773929610, −7.17532421098191671736427807454, −6.43450645195586596506138754629, −4.95215921956646499679372637528, −4.03385550845430111617584640533, −2.65083657567708775859548676348, −1.37212631195397430665609023107, 1.17691523541677055112906520354, 2.75039315023373836259544244855, 4.10633010925431507774267738429, 4.81303927149786691370753272604, 6.24868033981457398905397510089, 6.62437609088973882567823088300, 8.365831583571020927369759159465, 9.124101404826730105560546041016, 9.741887747357988453087675934144, 10.10199770078974556456290445272

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