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Properties

Label 2-637-91.76-c1-0-36
Degree $2$
Conductor $637$
Sign $0.333 + 0.942i$
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  + (0.179 − 0.670i)2-s + (2.71 − 1.56i)3-s + (1.31 + 0.758i)4-s + (−0.0263 + 0.0263i)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 + (−1.09 − 0.292i)11-s + 4.75·12-s + (−3.58 + 0.400i)13-s + (−0.0301 + 0.112i)15-s + (0.669 + 1.16i)16-s + (−3.20 + 5.54i)17-s + (−3.33 − 3.33i)18-s + (−0.954 − 3.56i)19-s + ⋯
L(s)  = 1  + (0.127 − 0.474i)2-s + (1.56 − 0.903i)3-s + (0.657 + 0.379i)4-s + (−0.0117 + 0.0117i)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 + (−0.328 − 0.0880i)11-s + 1.37·12-s + (−0.993 + 0.111i)13-s + (−0.00778 + 0.0290i)15-s + (0.167 + 0.290i)16-s + (−0.776 + 1.34i)17-s + (−0.786 − 0.786i)18-s + (−0.218 − 0.816i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.333 + 0.942i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.333 + 0.942i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $0.333 + 0.942i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (440, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ 0.333 + 0.942i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.42914 - 1.71738i\)
\(L(\frac12)\) \(\approx\) \(2.42914 - 1.71738i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.179 + 0.670i)T + (-1.73 - i)T^{2} \)
3 \( 1 + (-2.71 + 1.56i)T + (1.5 - 2.59i)T^{2} \)
5 \( 1 + (0.0263 - 0.0263i)T - 5iT^{2} \)
11 \( 1 + (1.09 + 0.292i)T + (9.52 + 5.5i)T^{2} \)
17 \( 1 + (3.20 - 5.54i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.954 + 3.56i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (2.41 - 1.39i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.84 - 3.20i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-1.93 + 1.93i)T - 31iT^{2} \)
37 \( 1 + (-5.39 - 1.44i)T + (32.0 + 18.5i)T^{2} \)
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 \( 1 + (3.97 + 3.97i)T + 47iT^{2} \)
53 \( 1 - 0.591T + 53T^{2} \)
59 \( 1 + (10.8 - 2.91i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (-1.18 - 0.686i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.95 + 7.28i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + (9.88 - 2.64i)T + (61.4 - 35.5i)T^{2} \)
73 \( 1 + (-1.93 - 1.93i)T + 73iT^{2} \)
79 \( 1 - 3.26T + 79T^{2} \)
83 \( 1 + (4.92 - 4.92i)T - 83iT^{2} \)
89 \( 1 + (-4.36 + 16.2i)T + (-77.0 - 44.5i)T^{2} \)
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

−10.39076820911781214597412003290, −9.426196958069766195987353959917, −8.529971963833272382199864065881, −7.75953712331244512112502722343, −7.13161045955907065423560200106, −6.29063858741433707067574541865, −4.39302843001332138213213564197, −3.31221674472705008928832425561, −2.48292483903124682460630103401, −1.65198132705362915829010918765, 2.22738296397403157556475530590, 2.82406333783635342848069388699, 4.33349760056645048474685065237, 4.99776425582558371817457716630, 6.36717481593374011428451231666, 7.52532506078928445357793626108, 7.989709092570740254654952786866, 9.018812328741600953334741604350, 9.937797798358991179157786341561, 10.32836785077113820992652086453

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