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Properties

Label 2-637-91.88-c1-0-23
Degree $2$
Conductor $637$
Sign $0.512 + 0.858i$
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  + (1.5 + 0.866i)2-s − 2·3-s + (0.5 + 0.866i)4-s + (−1.5 + 0.866i)5-s + (−3 − 1.73i)6-s − 1.73i·8-s + 9-s − 3·10-s + (−1 − 1.73i)12-s + (2.5 − 2.59i)13-s + (3 − 1.73i)15-s + (2.49 − 4.33i)16-s + (−1.5 − 2.59i)17-s + (1.5 + 0.866i)18-s − 3.46i·19-s + (−1.5 − 0.866i)20-s + ⋯
L(s)  = 1  + (1.06 + 0.612i)2-s − 1.15·3-s + (0.250 + 0.433i)4-s + (−0.670 + 0.387i)5-s + (−1.22 − 0.707i)6-s − 0.612i·8-s + 0.333·9-s − 0.948·10-s + (−0.288 − 0.500i)12-s + (0.693 − 0.720i)13-s + (0.774 − 0.447i)15-s + (0.624 − 1.08i)16-s + (−0.363 − 0.630i)17-s + (0.353 + 0.204i)18-s − 0.794i·19-s + (−0.335 − 0.193i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.512 + 0.858i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.512 + 0.858i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $0.512 + 0.858i$
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: \((2,\ 637,\ (\ :1/2),\ 0.512 + 0.858i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.924169 - 0.524544i\)
\(L(\frac12)\) \(\approx\) \(0.924169 - 0.524544i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.5 + 2.59i)T \)
good2 \( 1 + (-1.5 - 0.866i)T + (1 + 1.73i)T^{2} \)
3 \( 1 + 2T + 3T^{2} \)
5 \( 1 + (1.5 - 0.866i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 - 11T^{2} \)
17 \( 1 + (1.5 + 2.59i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + 3.46iT - 19T^{2} \)
23 \( 1 + (-3 + 5.19i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.5 + 2.59i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (3 + 1.73i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (7.5 + 4.33i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-4.5 + 2.59i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (4 - 6.92i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (3 - 1.73i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-1.5 + 2.59i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-6 + 3.46i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + T + 61T^{2} \)
67 \( 1 - 3.46iT - 67T^{2} \)
71 \( 1 + (-3 - 1.73i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (1.5 + 0.866i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (2 + 3.46i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 13.8iT - 83T^{2} \)
89 \( 1 + (6 + 3.46i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (6 + 3.46i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.93638211188093199515334930785, −9.733508496049652972552692847022, −8.505161975160933598036068709853, −7.26709058866778054975839607403, −6.68201826740585785137785132156, −5.77344741386487776485095227481, −5.08996316171099503409157474381, −4.16319865780015139198901909489, −3.05364108688510757743403718478, −0.48842600497963476640525270221, 1.62686852176340153291936467398, 3.41830582243927935744408857356, 4.18916674648145817787085134914, 5.13663646496709187077605882939, 5.84064458120069773276357046879, 6.85122568022402587135816121391, 8.153428390491766762025414323547, 8.925839659442933591879481987419, 10.40199763930213594460594816242, 11.08060364416739305060101066546

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