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Properties

Label 2-945-9.4-c1-0-18
Degree $2$
Conductor $945$
Sign $0.999 + 0.0364i$
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  + (0.875 + 1.51i)2-s + (−0.532 + 0.922i)4-s + (0.5 − 0.866i)5-s + (−0.5 − 0.866i)7-s + 1.63·8-s + 1.75·10-s + (−3.05 − 5.29i)11-s + (2.90 − 5.03i)13-s + (0.875 − 1.51i)14-s + (2.49 + 4.32i)16-s − 4.73·17-s + 3.68·19-s + (0.532 + 0.922i)20-s + (5.34 − 9.26i)22-s + (−0.5 + 0.866i)23-s + ⋯
L(s)  = 1  + (0.619 + 1.07i)2-s + (−0.266 + 0.461i)4-s + (0.223 − 0.387i)5-s + (−0.188 − 0.327i)7-s + 0.578·8-s + 0.553·10-s + (−0.921 − 1.59i)11-s + (0.806 − 1.39i)13-s + (0.233 − 0.405i)14-s + (0.624 + 1.08i)16-s − 1.14·17-s + 0.844·19-s + (0.119 + 0.206i)20-s + (1.14 − 1.97i)22-s + (−0.104 + 0.180i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 945 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0364i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 945 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0364i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(945\)    =    \(3^{3} \cdot 5 \cdot 7\)
Sign: $0.999 + 0.0364i$
Analytic conductor: \(7.54586\)
Root analytic conductor: \(2.74697\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{945} (631, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 945,\ (\ :1/2),\ 0.999 + 0.0364i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.23263 - 0.0407206i\)
\(L(\frac12)\) \(\approx\) \(2.23263 - 0.0407206i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 + 0.866i)T \)
7 \( 1 + (0.5 + 0.866i)T \)
good2 \( 1 + (-0.875 - 1.51i)T + (-1 + 1.73i)T^{2} \)
11 \( 1 + (3.05 + 5.29i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-2.90 + 5.03i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 4.73T + 17T^{2} \)
19 \( 1 - 3.68T + 19T^{2} \)
23 \( 1 + (0.5 - 0.866i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.839 + 1.45i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-3.47 + 6.02i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 4.49T + 37T^{2} \)
41 \( 1 + (1.98 - 3.44i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-5.66 - 9.80i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-2.08 - 3.61i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 8.30T + 53T^{2} \)
59 \( 1 + (-0.418 + 0.724i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4.62 - 8.01i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.25 - 3.91i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 10.3T + 71T^{2} \)
73 \( 1 - 12.2T + 73T^{2} \)
79 \( 1 + (4.41 + 7.64i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (3.17 + 5.49i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 9.66T + 89T^{2} \)
97 \( 1 + (-4.08 - 7.07i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.17166410939013298285320987930, −8.896687176569488634978874608495, −8.101703940467279360017653318799, −7.56581160812274059698776242851, −6.28423276305442560997735133474, −5.81755981294530433747767288495, −5.09075998134011300442282607294, −3.95845787734237625301911712650, −2.85009447498972041655227583462, −0.882093424056722657037204607774, 1.82637242397478844147521301162, 2.43510889142825396338825416850, 3.65697808582118392504799807701, 4.54424463037584916557707112325, 5.36849323393739787855955424484, 6.79604454002312810087308867042, 7.24286190526656145851303892927, 8.595197184723704254637955537370, 9.481468087781869779918085013286, 10.32967851249963920832372747813

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