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Properties

Label 2-945-7.4-c1-0-18
Degree $2$
Conductor $945$
Sign $0.0465 + 0.998i$
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  + (−0.904 − 1.56i)2-s + (−0.635 + 1.10i)4-s + (0.5 + 0.866i)5-s + (−2.62 + 0.290i)7-s − 1.31·8-s + (0.904 − 1.56i)10-s + (−0.546 + 0.946i)11-s + 5.71·13-s + (2.83 + 3.85i)14-s + (2.46 + 4.26i)16-s + (2.20 − 3.81i)17-s + (3.14 + 5.45i)19-s − 1.27·20-s + 1.97·22-s + (−3.09 − 5.36i)23-s + ⋯
L(s)  = 1  + (−0.639 − 1.10i)2-s + (−0.317 + 0.550i)4-s + (0.223 + 0.387i)5-s + (−0.993 + 0.109i)7-s − 0.466·8-s + (0.285 − 0.495i)10-s + (−0.164 + 0.285i)11-s + 1.58·13-s + (0.757 + 1.03i)14-s + (0.615 + 1.06i)16-s + (0.534 − 0.924i)17-s + (0.722 + 1.25i)19-s − 0.284·20-s + 0.421·22-s + (−0.646 − 1.11i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 945 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0465 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 945 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0465 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(945\)    =    \(3^{3} \cdot 5 \cdot 7\)
Sign: $0.0465 + 0.998i$
Analytic conductor: \(7.54586\)
Root analytic conductor: \(2.74697\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{945} (676, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 945,\ (\ :1/2),\ 0.0465 + 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.755240 - 0.720875i\)
\(L(\frac12)\) \(\approx\) \(0.755240 - 0.720875i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.62 - 0.290i)T \)
good2 \( 1 + (0.904 + 1.56i)T + (-1 + 1.73i)T^{2} \)
11 \( 1 + (0.546 - 0.946i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 5.71T + 13T^{2} \)
17 \( 1 + (-2.20 + 3.81i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.14 - 5.45i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (3.09 + 5.36i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 3.61T + 29T^{2} \)
31 \( 1 + (-3.87 + 6.71i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-0.0888 - 0.153i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 8.46T + 41T^{2} \)
43 \( 1 - 6.47T + 43T^{2} \)
47 \( 1 + (-4.28 - 7.41i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-5.69 + 9.86i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (0.364 - 0.632i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.96 + 5.13i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.607 + 1.05i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 6.19T + 71T^{2} \)
73 \( 1 + (-5.45 + 9.44i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (1.01 + 1.76i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 9.07T + 83T^{2} \)
89 \( 1 + (-8.05 - 13.9i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 11.6T + 97T^{2} \)
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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

−9.814868873261009680929510996175, −9.438971033168476912981490031225, −8.444913619407480011683655950552, −7.48942079062972675994570756840, −6.16736727854731808627826124876, −5.86024317136843045538758160376, −3.99560760725917425770425956650, −3.18056223345639312549703495242, −2.23991930026569538083003112377, −0.824368857089405773705417130165, 0.989053617222321721525748960553, 2.99294786452500594314000768653, 3.95078168375975234301648354878, 5.66077333773627826160136807953, 5.92260861619842520097541854264, 6.93411490264367305620121901356, 7.68590890510406112522818486340, 8.676723976711107439485499755386, 9.085649918165703554512474693598, 9.944497256567045545894183571523

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