Properties

Label 2-637-7.4-c1-0-29
Degree $2$
Conductor $637$
Sign $0.605 + 0.795i$
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)3-s + (1 − 1.73i)4-s + (1.5 + 2.59i)5-s + (−0.499 − 0.866i)9-s + (−1.99 − 3.46i)12-s + 13-s + 6·15-s + (−1.99 − 3.46i)16-s + (3 − 5.19i)17-s + (3.5 + 6.06i)19-s + 6·20-s + (−1.5 − 2.59i)23-s + (−2 + 3.46i)25-s + 4.00·27-s − 9·29-s + ⋯
L(s)  = 1  + (0.577 − 0.999i)3-s + (0.5 − 0.866i)4-s + (0.670 + 1.16i)5-s + (−0.166 − 0.288i)9-s + (−0.577 − 0.999i)12-s + 0.277·13-s + 1.54·15-s + (−0.499 − 0.866i)16-s + (0.727 − 1.26i)17-s + (0.802 + 1.39i)19-s + 1.34·20-s + (−0.312 − 0.541i)23-s + (−0.400 + 0.692i)25-s + 0.769·27-s − 1.67·29-s + ⋯

Functional equation

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

Invariants

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

Particular Values

\(L(1)\) \(\approx\) \(2.04520 - 1.01386i\)
\(L(\frac12)\) \(\approx\) \(2.04520 - 1.01386i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1 + 1.73i)T^{2} \)
3 \( 1 + (-1 + 1.73i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (-1.5 - 2.59i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (-3 + 5.19i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.5 - 6.06i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.5 + 2.59i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 9T + 29T^{2} \)
31 \( 1 + (2.5 - 4.33i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (1 + 1.73i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 6T + 41T^{2} \)
43 \( 1 + T + 43T^{2} \)
47 \( 1 + (1.5 + 2.59i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-4.5 + 7.79i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-5 - 8.66i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (7 - 12.1i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 6T + 71T^{2} \)
73 \( 1 + (5.5 - 9.52i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-0.5 - 0.866i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 3T + 83T^{2} \)
89 \( 1 + (7.5 + 12.9i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + T + 97T^{2} \)
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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

−10.20178575872346732636842172104, −9.943476455264955985501598289319, −8.666662431911456873433076355555, −7.32382781730063247368061341758, −7.15422656240831436384898795626, −6.05956903679745416831384629871, −5.34970223126175432920954939460, −3.33655919731409294451986272758, −2.35575860272932616176503063076, −1.45239219827050149743360769574, 1.72524943218544239780109093382, 3.20996845857271698451291459010, 4.00330740096828896103718494942, 5.05728078844996327807484628377, 6.06474884243324355243380018483, 7.40020227169092827934891774212, 8.307352463725448282309612326081, 9.099520615524474934016293733985, 9.547652114237072681773877091954, 10.60320733949262905995285434806

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