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Properties

Label 2-637-91.88-c1-0-33
Degree $2$
Conductor $637$
Sign $0.243 + 0.969i$
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  + (1.16 + 0.672i)2-s − 2.05·3-s + (−0.0951 − 0.164i)4-s + (3.08 − 1.78i)5-s + (−2.38 − 1.37i)6-s − 2.94i·8-s + 1.20·9-s + 4.79·10-s + 1.27i·11-s + (0.195 + 0.337i)12-s + (−3.57 + 0.474i)13-s + (−6.33 + 3.65i)15-s + (1.79 − 3.10i)16-s + (−3.86 − 6.70i)17-s + (1.40 + 0.809i)18-s − 0.943i·19-s + ⋯
L(s)  = 1  + (0.823 + 0.475i)2-s − 1.18·3-s + (−0.0475 − 0.0824i)4-s + (1.38 − 0.797i)5-s + (−0.975 − 0.562i)6-s − 1.04i·8-s + 0.400·9-s + 1.51·10-s + 0.385i·11-s + (0.0563 + 0.0975i)12-s + (−0.991 + 0.131i)13-s + (−1.63 + 0.944i)15-s + (0.447 − 0.775i)16-s + (−0.938 − 1.62i)17-s + (0.330 + 0.190i)18-s − 0.216i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.243 + 0.969i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.243 + 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $0.243 + 0.969i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ 0.243 + 0.969i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.15620 - 0.901876i\)
\(L(\frac12)\) \(\approx\) \(1.15620 - 0.901876i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.57 - 0.474i)T \)
good2 \( 1 + (-1.16 - 0.672i)T + (1 + 1.73i)T^{2} \)
3 \( 1 + 2.05T + 3T^{2} \)
5 \( 1 + (-3.08 + 1.78i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 - 1.27iT - 11T^{2} \)
17 \( 1 + (3.86 + 6.70i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + 0.943iT - 19T^{2} \)
23 \( 1 + (-0.823 + 1.42i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (2.02 + 3.50i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-4.46 - 2.57i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.914 - 0.528i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-3.63 + 2.09i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-1.91 + 3.31i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-0.774 + 0.447i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-0.0399 + 0.0692i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (9.68 - 5.59i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 - 7.62T + 61T^{2} \)
67 \( 1 + 6.32iT - 67T^{2} \)
71 \( 1 + (-9.89 - 5.71i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (-0.658 - 0.380i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-1.42 - 2.47i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 2.32iT - 83T^{2} \)
89 \( 1 + (6.56 + 3.78i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-0.414 - 0.239i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.28457533070309937419447774516, −9.624570991930731399097095214118, −8.970374887560245344255189017860, −7.20587665881694244015913487888, −6.46703438038422022781281525496, −5.69059342131290068444067460993, −4.94291716954783603282746619170, −4.60896054312748269280693885404, −2.41761650688533262407628021225, −0.69385510279080677710331791503, 1.96979330899060420768622882537, 3.00300509953736237249361608329, 4.38030100800440185510944897422, 5.39846043234999822088811911935, 5.99315171047434209716095887674, 6.72264158121706525411475682281, 8.100928732483274538657366512353, 9.293054255689326891674362969943, 10.32398586389184239512651203110, 10.89800825262259833716110413130

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