Properties

Label 2-637-7.4-c1-0-4
Degree $2$
Conductor $637$
Sign $0.386 - 0.922i$
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  + (−1.36 − 2.36i)2-s + (−0.673 + 1.16i)3-s + (−2.71 + 4.69i)4-s + (1.09 + 1.89i)5-s + 3.66·6-s + 9.33·8-s + (0.593 + 1.02i)9-s + (2.98 − 5.16i)10-s + (0.524 − 0.907i)11-s + (−3.65 − 6.32i)12-s − 13-s − 2.94·15-s + (−7.29 − 12.6i)16-s + (−2.64 + 4.58i)17-s + (1.61 − 2.80i)18-s + (0.378 + 0.655i)19-s + ⋯
L(s)  = 1  + (−0.963 − 1.66i)2-s + (−0.388 + 0.673i)3-s + (−1.35 + 2.34i)4-s + (0.489 + 0.847i)5-s + 1.49·6-s + 3.30·8-s + (0.197 + 0.342i)9-s + (0.942 − 1.63i)10-s + (0.158 − 0.273i)11-s + (−1.05 − 1.82i)12-s − 0.277·13-s − 0.760·15-s + (−1.82 − 3.16i)16-s + (−0.641 + 1.11i)17-s + (0.381 − 0.660i)18-s + (0.0868 + 0.150i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.386 - 0.922i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.386 - 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $0.386 - 0.922i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (508, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ 0.386 - 0.922i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.447657 + 0.297773i\)
\(L(\frac12)\) \(\approx\) \(0.447657 + 0.297773i\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
good2 \( 1 + (1.36 + 2.36i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + (0.673 - 1.16i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (-1.09 - 1.89i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-0.524 + 0.907i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (2.64 - 4.58i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.378 - 0.655i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.326 + 0.566i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 3.10T + 29T^{2} \)
31 \( 1 + (-0.513 + 0.890i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-5.44 - 9.43i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 7.32T + 41T^{2} \)
43 \( 1 - 0.887T + 43T^{2} \)
47 \( 1 + (-1.16 - 2.02i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (2.44 - 4.23i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (0.524 - 0.907i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (6.24 + 10.8i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.23 - 3.87i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 6.60T + 71T^{2} \)
73 \( 1 + (4.14 - 7.17i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (1.07 + 1.85i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 6.66T + 83T^{2} \)
89 \( 1 + (2.88 + 4.99i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 2.88T + 97T^{2} \)
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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

−10.61390199376315823979350932954, −10.15075693554016048993210311364, −9.462342276217135799663855220641, −8.485839190671764509081686824462, −7.60001356097896440247744695087, −6.30154255488886456688419677726, −4.77188941934893425910890704959, −3.84674534042961743953037604217, −2.76663578094692919485230202755, −1.70217183328077112493949348873, 0.44028680681346260899922833916, 1.66182544439942554344559947374, 4.47938349446214616066162659524, 5.36499972964153695003218330480, 6.11116971821085029702495207443, 7.05665322126163281527216324187, 7.48726941882056991968612698939, 8.684527519997072663864102586542, 9.306933854465863408556833824703, 9.832998236907154685262966120607

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