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Properties

Label 2-945-15.2-c1-0-31
Degree $2$
Conductor $945$
Sign $0.240 + 0.970i$
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  + (−0.885 + 0.885i)2-s + 0.430i·4-s + (−0.684 + 2.12i)5-s + (−0.707 − 0.707i)7-s + (−2.15 − 2.15i)8-s + (−1.28 − 2.49i)10-s + 0.102i·11-s + (−0.320 + 0.320i)13-s + 1.25·14-s + 2.95·16-s + (−4.37 + 4.37i)17-s − 6.18i·19-s + (−0.915 − 0.294i)20-s + (−0.0910 − 0.0910i)22-s + (−1.19 − 1.19i)23-s + ⋯
L(s)  = 1  + (−0.626 + 0.626i)2-s + 0.215i·4-s + (−0.305 + 0.952i)5-s + (−0.267 − 0.267i)7-s + (−0.761 − 0.761i)8-s + (−0.404 − 0.788i)10-s + 0.0309i·11-s + (−0.0889 + 0.0889i)13-s + 0.334·14-s + 0.738·16-s + (−1.06 + 1.06i)17-s − 1.41i·19-s + (−0.204 − 0.0657i)20-s + (−0.0194 − 0.0194i)22-s + (−0.250 − 0.250i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 945 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.240 + 0.970i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 945 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.240 + 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(945\)    =    \(3^{3} \cdot 5 \cdot 7\)
Sign: $0.240 + 0.970i$
Analytic conductor: \(7.54586\)
Root analytic conductor: \(2.74697\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{945} (512, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 945,\ (\ :1/2),\ 0.240 + 0.970i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0926726 - 0.0725290i\)
\(L(\frac12)\) \(\approx\) \(0.0926726 - 0.0725290i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.684 - 2.12i)T \)
7 \( 1 + (0.707 + 0.707i)T \)
good2 \( 1 + (0.885 - 0.885i)T - 2iT^{2} \)
11 \( 1 - 0.102iT - 11T^{2} \)
13 \( 1 + (0.320 - 0.320i)T - 13iT^{2} \)
17 \( 1 + (4.37 - 4.37i)T - 17iT^{2} \)
19 \( 1 + 6.18iT - 19T^{2} \)
23 \( 1 + (1.19 + 1.19i)T + 23iT^{2} \)
29 \( 1 + 6.40T + 29T^{2} \)
31 \( 1 - 6.61T + 31T^{2} \)
37 \( 1 + (3.33 + 3.33i)T + 37iT^{2} \)
41 \( 1 + 6.60iT - 41T^{2} \)
43 \( 1 + (2.74 - 2.74i)T - 43iT^{2} \)
47 \( 1 + (1.77 - 1.77i)T - 47iT^{2} \)
53 \( 1 + (-2.88 - 2.88i)T + 53iT^{2} \)
59 \( 1 - 12.8T + 59T^{2} \)
61 \( 1 + 3.25T + 61T^{2} \)
67 \( 1 + (1.55 + 1.55i)T + 67iT^{2} \)
71 \( 1 + 3.88iT - 71T^{2} \)
73 \( 1 + (-9.24 + 9.24i)T - 73iT^{2} \)
79 \( 1 + 2.10iT - 79T^{2} \)
83 \( 1 + (8.52 + 8.52i)T + 83iT^{2} \)
89 \( 1 + 12.6T + 89T^{2} \)
97 \( 1 + (10.4 + 10.4i)T + 97iT^{2} \)
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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

−9.741753991103146315652584174743, −8.886180448251232429742442012185, −8.165824175278718840983684762652, −7.20162785652136136539289368640, −6.78073021050681246657360591794, −5.96745942751127279927631239207, −4.37113643534976633766720863078, −3.49802646574926997655837238716, −2.39206843689335698471638706969, −0.06921564646274742668403609693, 1.34641963890309612753194647213, 2.50728924151822778312279950509, 3.85764380511449581839462705758, 5.04614304192728942349452844593, 5.74815537664817105507546940151, 6.87270280593312280525283647480, 8.164866932424268513653874652926, 8.608582519876064166487199481393, 9.646798282964287883781374027921, 9.878164516629314638500512660531

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