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Properties

Label 2-637-91.51-c1-0-19
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: $\chi_{637} (324, \cdot )$
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

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

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