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Properties

Label 2-637-91.5-c1-0-30
Degree $2$
Conductor $637$
Sign $0.271 + 0.962i$
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  + (−0.147 − 0.550i)2-s + (1.06 − 0.616i)3-s + (1.45 − 0.837i)4-s + (1.41 − 0.377i)5-s + (−0.496 − 0.496i)6-s + (−1.48 − 1.48i)8-s + (−0.740 + 1.28i)9-s + (−0.416 − 0.720i)10-s + (−0.218 + 0.815i)11-s + (1.03 − 1.78i)12-s + (3.59 + 0.296i)13-s + (1.27 − 1.27i)15-s + (1.07 − 1.86i)16-s + (−3.67 − 6.36i)17-s + (0.815 + 0.218i)18-s + (4.90 − 1.31i)19-s + ⋯
L(s)  = 1  + (−0.104 − 0.389i)2-s + (0.616 − 0.355i)3-s + (0.725 − 0.418i)4-s + (0.630 − 0.168i)5-s + (−0.202 − 0.202i)6-s + (−0.523 − 0.523i)8-s + (−0.246 + 0.427i)9-s + (−0.131 − 0.227i)10-s + (−0.0658 + 0.245i)11-s + (0.297 − 0.516i)12-s + (0.996 + 0.0822i)13-s + (0.328 − 0.328i)15-s + (0.269 − 0.466i)16-s + (−0.891 − 1.54i)17-s + (0.192 + 0.0515i)18-s + (1.12 − 0.301i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.271 + 0.962i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.271 + 0.962i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $0.271 + 0.962i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (460, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ 0.271 + 0.962i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.76829 - 1.33834i\)
\(L(\frac12)\) \(\approx\) \(1.76829 - 1.33834i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.59 - 0.296i)T \)
good2 \( 1 + (0.147 + 0.550i)T + (-1.73 + i)T^{2} \)
3 \( 1 + (-1.06 + 0.616i)T + (1.5 - 2.59i)T^{2} \)
5 \( 1 + (-1.41 + 0.377i)T + (4.33 - 2.5i)T^{2} \)
11 \( 1 + (0.218 - 0.815i)T + (-9.52 - 5.5i)T^{2} \)
17 \( 1 + (3.67 + 6.36i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-4.90 + 1.31i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + (-3.84 - 2.22i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 3.54T + 29T^{2} \)
31 \( 1 + (-0.465 + 1.73i)T + (-26.8 - 15.5i)T^{2} \)
37 \( 1 + (3.93 - 1.05i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (1.23 + 1.23i)T + 41iT^{2} \)
43 \( 1 - 8.66iT - 43T^{2} \)
47 \( 1 + (0.923 + 3.44i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (-4.89 - 8.48i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (1.48 + 0.398i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (-6.15 - 3.55i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (11.9 + 3.20i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + (1.46 - 1.46i)T - 71iT^{2} \)
73 \( 1 + (0.141 + 0.0380i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (-2.39 + 4.15i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (12.3 + 12.3i)T + 83iT^{2} \)
89 \( 1 + (-2.52 - 9.41i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (-6.05 - 6.05i)T + 97iT^{2} \)
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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

−10.44272915258544931467351953100, −9.381725917974207005919236762509, −9.002809081345160171029881581279, −7.61832135291675708092426833615, −6.99689556209936797175561822801, −5.86912271250187904884604597645, −5.02126296989521478348086243154, −3.26427014351848421218636088735, −2.38931257493603453861336962638, −1.33611947143826059772421872025, 1.88160712087273260665851366575, 3.11464544855516080585124783924, 3.89121907387687054112632855311, 5.65919850085558641733540212556, 6.26406687867813037457096235103, 7.17821081807058484832439334582, 8.472140007085335067863641369717, 8.650447882240502098610233673115, 9.833910246331449497866136219840, 10.73413901700885884760787132425

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