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Properties

Label 2-945-105.59-c1-0-60
Degree $2$
Conductor $945$
Sign $-0.204 - 0.978i$
Analytic cond. $7.54586$
Root an. cond. $2.74697$
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  + (−0.921 − 1.59i)2-s + (−0.699 + 1.21i)4-s + (−0.939 − 2.02i)5-s + (0.918 − 2.48i)7-s − 1.10·8-s + (−2.37 + 3.37i)10-s + (0.962 + 0.555i)11-s − 0.321·13-s + (−4.80 + 0.820i)14-s + (2.42 + 4.19i)16-s + (−5.55 − 3.21i)17-s + (−0.813 + 0.469i)19-s + (3.11 + 0.280i)20-s − 2.04i·22-s + (−2.36 − 4.09i)23-s + ⋯
L(s)  = 1  + (−0.651 − 1.12i)2-s + (−0.349 + 0.605i)4-s + (−0.420 − 0.907i)5-s + (0.347 − 0.937i)7-s − 0.392·8-s + (−0.750 + 1.06i)10-s + (0.290 + 0.167i)11-s − 0.0890·13-s + (−1.28 + 0.219i)14-s + (0.605 + 1.04i)16-s + (−1.34 − 0.778i)17-s + (−0.186 + 0.107i)19-s + (0.696 + 0.0628i)20-s − 0.436i·22-s + (−0.493 − 0.854i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 945 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.204 - 0.978i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 945 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.204 - 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(945\)    =    \(3^{3} \cdot 5 \cdot 7\)
Sign: $-0.204 - 0.978i$
Analytic conductor: \(7.54586\)
Root analytic conductor: \(2.74697\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{945} (269, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 945,\ (\ :1/2),\ -0.204 - 0.978i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.303572 + 0.373517i\)
\(L(\frac12)\) \(\approx\) \(0.303572 + 0.373517i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 + (0.939 + 2.02i)T \)
7 \( 1 + (-0.918 + 2.48i)T \)
good2 \( 1 + (0.921 + 1.59i)T + (-1 + 1.73i)T^{2} \)
11 \( 1 + (-0.962 - 0.555i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + 0.321T + 13T^{2} \)
17 \( 1 + (5.55 + 3.21i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.813 - 0.469i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (2.36 + 4.09i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 1.65iT - 29T^{2} \)
31 \( 1 + (2.30 + 1.33i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-6.89 + 3.98i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 - 9.11T + 41T^{2} \)
43 \( 1 - 1.80iT - 43T^{2} \)
47 \( 1 + (4.37 - 2.52i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (1.42 - 2.46i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (0.493 - 0.855i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (8.13 - 4.69i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (8.27 + 4.77i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 13.6iT - 71T^{2} \)
73 \( 1 + (6.46 - 11.2i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-2.91 - 5.05i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 8.94iT - 83T^{2} \)
89 \( 1 + (6.72 + 11.6i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 9.19T + 97T^{2} \)
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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

−9.461513031353964135410673691256, −8.890107635411194227673324306104, −8.044235281063680248361947579454, −7.14184995144831094486262470468, −5.96882317400765249291537965089, −4.55565112286045744236136619521, −4.05859374203829638354151347966, −2.61316028395264067499538759717, −1.40815864233533434550369507301, −0.28389141740119444917632075337, 2.18421404909901075182060259292, 3.37016204519049897693514691976, 4.66751314813385489665041118283, 6.09263139879796398307204071111, 6.27166733487202371128552438775, 7.42511288261628736235459466783, 7.995412203058163321637851740042, 8.836498839487918963634166368336, 9.443708200261180293896821762074, 10.56351975111372213069122035026

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