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Properties

Label 2-637-91.25-c1-0-22
Degree $2$
Conductor $637$
Sign $0.795 - 0.605i$
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 + 1.73i)4-s + (3.12 − 1.80i)5-s + (1.5 + 2.59i)9-s + 3.60i·13-s + (−1.99 − 3.46i)16-s + (3.12 − 1.80i)19-s + 7.21i·20-s + (−0.5 − 0.866i)23-s + (4 − 6.92i)25-s − 5·29-s + (9.36 + 5.40i)31-s − 6·36-s + 7.21i·41-s + 9·43-s + (9.36 + 5.40i)45-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)4-s + (1.39 − 0.806i)5-s + (0.5 + 0.866i)9-s + 0.999i·13-s + (−0.499 − 0.866i)16-s + (0.716 − 0.413i)19-s + 1.61i·20-s + (−0.104 − 0.180i)23-s + (0.800 − 1.38i)25-s − 0.928·29-s + (1.68 + 0.971i)31-s − 36-s + 1.12i·41-s + 1.37·43-s + (1.39 + 0.806i)45-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.795 - 0.605i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.795 - 0.605i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $0.795 - 0.605i$
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: \((2,\ 637,\ (\ :1/2),\ 0.795 - 0.605i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.64532 + 0.554710i\)
\(L(\frac12)\) \(\approx\) \(1.64532 + 0.554710i\)
\(L(\frac{3}{2})\) 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 \( 1 - 3.60iT \)
good2 \( 1 + (1 - 1.73i)T^{2} \)
3 \( 1 + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (-3.12 + 1.80i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.12 + 1.80i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.5 + 0.866i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 5T + 29T^{2} \)
31 \( 1 + (-9.36 - 5.40i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (18.5 - 32.0i)T^{2} \)
41 \( 1 - 7.21iT - 41T^{2} \)
43 \( 1 - 9T + 43T^{2} \)
47 \( 1 + (-3.12 + 1.80i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (5.5 - 9.52i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (12.4 + 7.21i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 + (-9.36 - 5.40i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (7.5 + 12.9i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 18.0iT - 83T^{2} \)
89 \( 1 + (-3.12 + 1.80i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 18.0iT - 97T^{2} \)
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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

−10.50405602603811162999655128829, −9.516988480004114563912584675238, −9.130475613100103490862390410365, −8.154838009127354840871799790026, −7.22736359153348885589483677072, −6.09822562860099551616449803030, −4.94003789492775099633991488438, −4.42546371503590112072054326494, −2.77698270061114537509774701645, −1.54817598131703018215100034517, 1.13070580468281495959922588167, 2.50081488287568496765062229290, 3.85090884608429443645664173624, 5.28596404846057207592207687406, 5.95291657749220701817875402989, 6.61661008332818910175282622763, 7.80379735615871132070890014767, 9.225974662347190157997786463490, 9.668473458891537659779895855166, 10.30474533319827302725125635431

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