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Properties

Label 2-637-91.51-c1-0-35
Degree $2$
Conductor $637$
Sign $-0.582 + 0.812i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.582 + 0.812i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.582 + 0.812i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $-0.582 + 0.812i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (324, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ -0.582 + 0.812i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.300595 - 0.585614i\)
\(L(\frac12)\) \(\approx\) \(0.300595 - 0.585614i\)
\(L(\frac{3}{2})\) 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

−10.35480974635449274105900085570, −9.465229015603155115751951723135, −8.714992796396993180139014659430, −7.70543749117742919269912018241, −6.73003553144341379312752362201, −5.20971478349493913085836989572, −4.86399186909377756230108867818, −3.84159167017593889141585022258, −2.47392773978550753491210806465, −0.30235858155446112711627558615, 2.17144695090236116097738263453, 3.12617215755457700523872645177, 4.22732226014662132215232713365, 5.25957487722664403986210892462, 6.70410524922246319598723822706, 7.55473694366442207734319471491, 8.216850529022441015739741344836, 8.700769887483191832183908749741, 10.13471162000169287092675112967, 11.00010203158950866682952862071

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