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Properties

Label 2-637-91.45-c1-0-32
Degree $2$
Conductor $637$
Sign $0.740 + 0.671i$
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  + (0.490 − 0.490i)2-s + (2.71 − 1.56i)3-s + 1.51i·4-s + (0.00962 − 0.0359i)5-s + (0.562 − 2.09i)6-s + (1.72 + 1.72i)8-s + (3.39 − 5.88i)9-s + (−0.0129 − 0.0223i)10-s + (0.292 − 1.09i)11-s + (2.37 + 4.11i)12-s + (3.58 + 0.400i)13-s + (−0.0301 − 0.112i)15-s − 1.33·16-s − 6.40·17-s + (−1.22 − 4.55i)18-s + (−3.56 + 0.954i)19-s + ⋯
L(s)  = 1  + (0.347 − 0.347i)2-s + (1.56 − 0.903i)3-s + 0.758i·4-s + (0.00430 − 0.0160i)5-s + (0.229 − 0.857i)6-s + (0.610 + 0.610i)8-s + (1.13 − 1.96i)9-s + (−0.00408 − 0.00707i)10-s + (0.0880 − 0.328i)11-s + (0.685 + 1.18i)12-s + (0.993 + 0.111i)13-s + (−0.00778 − 0.0290i)15-s − 0.334·16-s − 1.55·17-s + (−0.287 − 1.07i)18-s + (−0.816 + 0.218i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.740 + 0.671i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.740 + 0.671i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $0.740 + 0.671i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (227, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ 0.740 + 0.671i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.79259 - 1.07807i\)
\(L(\frac12)\) \(\approx\) \(2.79259 - 1.07807i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.58 - 0.400i)T \)
good2 \( 1 + (-0.490 + 0.490i)T - 2iT^{2} \)
3 \( 1 + (-2.71 + 1.56i)T + (1.5 - 2.59i)T^{2} \)
5 \( 1 + (-0.00962 + 0.0359i)T + (-4.33 - 2.5i)T^{2} \)
11 \( 1 + (-0.292 + 1.09i)T + (-9.52 - 5.5i)T^{2} \)
17 \( 1 + 6.40T + 17T^{2} \)
19 \( 1 + (3.56 - 0.954i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 - 2.79iT - 23T^{2} \)
29 \( 1 + (-1.84 + 3.20i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-2.63 + 0.706i)T + (26.8 - 15.5i)T^{2} \)
37 \( 1 + (3.94 + 3.94i)T + 37iT^{2} \)
41 \( 1 + (-0.188 + 0.0505i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (-1.84 + 1.06i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (5.43 + 1.45i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (0.295 - 0.512i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (7.97 - 7.97i)T - 59iT^{2} \)
61 \( 1 + (-1.18 - 0.686i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (7.28 + 1.95i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + (9.88 + 2.64i)T + (61.4 + 35.5i)T^{2} \)
73 \( 1 + (0.707 + 2.64i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (1.63 + 2.82i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-4.92 - 4.92i)T + 83iT^{2} \)
89 \( 1 + (11.9 - 11.9i)T - 89iT^{2} \)
97 \( 1 + (-0.663 + 2.47i)T + (-84.0 - 48.5i)T^{2} \)
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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

−10.62420459128790706979106805395, −9.065297937705019779413062725662, −8.738040899240960295699233275860, −7.988821876744305382515902580114, −7.12145459158273149685573617502, −6.29488240954001314681199248300, −4.40933962796901388031819643538, −3.57650097859126907236341025540, −2.67585845799527786540024947889, −1.69378197478383060832631822436, 1.85153426720800804154011800218, 3.06132450885737441538343891748, 4.37223221796989179549214401503, 4.69653685636564879863173178249, 6.26220555181974063047818201793, 7.07117142988512271754587958763, 8.488544350309983719301890033574, 8.769894346492701219244203748210, 9.768344014778304185298646565642, 10.52665270118201654903765145856

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