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Properties

Label 2-740-37.10-c1-0-11
Degree $2$
Conductor $740$
Sign $-0.964 + 0.263i$
Analytic cond. $5.90892$
Root an. cond. $2.43082$
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  + (−0.795 − 1.37i)3-s + (−0.5 − 0.866i)5-s + (−1.53 − 2.66i)7-s + (0.233 − 0.405i)9-s + 3.41·11-s + (0.945 + 1.63i)13-s + (−0.795 + 1.37i)15-s + (1.96 − 3.40i)17-s + (−0.897 − 1.55i)19-s + (−2.44 + 4.23i)21-s − 9.16·23-s + (−0.499 + 0.866i)25-s − 5.51·27-s − 2.24·29-s − 3.62·31-s + ⋯
L(s)  = 1  + (−0.459 − 0.795i)3-s + (−0.223 − 0.387i)5-s + (−0.580 − 1.00i)7-s + (0.0779 − 0.135i)9-s + 1.03·11-s + (0.262 + 0.454i)13-s + (−0.205 + 0.355i)15-s + (0.476 − 0.826i)17-s + (−0.205 − 0.356i)19-s + (−0.533 + 0.923i)21-s − 1.91·23-s + (−0.0999 + 0.173i)25-s − 1.06·27-s − 0.416·29-s − 0.651·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 740 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.964 + 0.263i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 740 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.964 + 0.263i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(740\)    =    \(2^{2} \cdot 5 \cdot 37\)
Sign: $-0.964 + 0.263i$
Analytic conductor: \(5.90892\)
Root analytic conductor: \(2.43082\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{740} (121, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 740,\ (\ :1/2),\ -0.964 + 0.263i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.116525 - 0.868305i\)
\(L(\frac12)\) \(\approx\) \(0.116525 - 0.868305i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (0.5 + 0.866i)T \)
37 \( 1 + (6.03 - 0.726i)T \)
good3 \( 1 + (0.795 + 1.37i)T + (-1.5 + 2.59i)T^{2} \)
7 \( 1 + (1.53 + 2.66i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 - 3.41T + 11T^{2} \)
13 \( 1 + (-0.945 - 1.63i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-1.96 + 3.40i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.897 + 1.55i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + 9.16T + 23T^{2} \)
29 \( 1 + 2.24T + 29T^{2} \)
31 \( 1 + 3.62T + 31T^{2} \)
41 \( 1 + (-3.30 - 5.73i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 - 6.37T + 43T^{2} \)
47 \( 1 + 3.32T + 47T^{2} \)
53 \( 1 + (-1.41 + 2.44i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-5.10 + 8.84i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3.65 - 6.33i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (6.34 + 10.9i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (2.10 + 3.65i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 - 4.74T + 73T^{2} \)
79 \( 1 + (0.346 + 0.599i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (5.27 - 9.13i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (4.29 - 7.43i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 3.39T + 97T^{2} \)
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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

−9.831889728681321072147709672164, −9.295149622599141330534835337704, −8.078779595040155206689893931984, −7.15870658572098240319431379046, −6.62842089869566935741449931402, −5.73534460870094490418312611679, −4.29035070360743908165654835016, −3.59492729692665109397571206107, −1.68484272330559826601901895246, −0.48165199457674286313000900505, 2.02488443039844408422707875584, 3.55095316949643390058424379603, 4.19047944578596062096936876303, 5.73331650857921470579145636943, 5.94049333974058223377650033277, 7.25932479172603608274121459691, 8.324147776293514157761450730764, 9.188679905110070212884320513079, 10.05225482794268716223335756226, 10.59070227419161346410760473191

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