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Properties

Label 2-945-35.27-c1-0-55
Degree $2$
Conductor $945$
Sign $0.344 + 0.938i$
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  + (0.834 + 0.834i)2-s − 0.606i·4-s + (1.65 − 1.50i)5-s + (−2.61 − 0.411i)7-s + (2.17 − 2.17i)8-s + (2.63 + 0.121i)10-s − 5.36·11-s + (−0.211 − 0.211i)13-s + (−1.83 − 2.52i)14-s + 2.41·16-s + (4.64 − 4.64i)17-s + 4.27·19-s + (−0.914 − 1.00i)20-s + (−4.48 − 4.48i)22-s + (−4.44 + 4.44i)23-s + ⋯
L(s)  = 1  + (0.590 + 0.590i)2-s − 0.303i·4-s + (0.738 − 0.673i)5-s + (−0.987 − 0.155i)7-s + (0.769 − 0.769i)8-s + (0.833 + 0.0384i)10-s − 1.61·11-s + (−0.0585 − 0.0585i)13-s + (−0.491 − 0.674i)14-s + 0.604·16-s + (1.12 − 1.12i)17-s + 0.979·19-s + (−0.204 − 0.224i)20-s + (−0.955 − 0.955i)22-s + (−0.926 + 0.926i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 945 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.344 + 0.938i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 945 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.344 + 0.938i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(945\)    =    \(3^{3} \cdot 5 \cdot 7\)
Sign: $0.344 + 0.938i$
Analytic conductor: \(7.54586\)
Root analytic conductor: \(2.74697\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{945} (622, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 945,\ (\ :1/2),\ 0.344 + 0.938i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.55111 - 1.08346i\)
\(L(\frac12)\) \(\approx\) \(1.55111 - 1.08346i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.65 + 1.50i)T \)
7 \( 1 + (2.61 + 0.411i)T \)
good2 \( 1 + (-0.834 - 0.834i)T + 2iT^{2} \)
11 \( 1 + 5.36T + 11T^{2} \)
13 \( 1 + (0.211 + 0.211i)T + 13iT^{2} \)
17 \( 1 + (-4.64 + 4.64i)T - 17iT^{2} \)
19 \( 1 - 4.27T + 19T^{2} \)
23 \( 1 + (4.44 - 4.44i)T - 23iT^{2} \)
29 \( 1 + 4.03iT - 29T^{2} \)
31 \( 1 + 6.55iT - 31T^{2} \)
37 \( 1 + (4.85 + 4.85i)T + 37iT^{2} \)
41 \( 1 - 1.91iT - 41T^{2} \)
43 \( 1 + (-5.01 + 5.01i)T - 43iT^{2} \)
47 \( 1 + (7.26 - 7.26i)T - 47iT^{2} \)
53 \( 1 + (-1.40 + 1.40i)T - 53iT^{2} \)
59 \( 1 + 1.34T + 59T^{2} \)
61 \( 1 + 1.03iT - 61T^{2} \)
67 \( 1 + (-7.23 - 7.23i)T + 67iT^{2} \)
71 \( 1 - 2.92T + 71T^{2} \)
73 \( 1 + (-9.72 - 9.72i)T + 73iT^{2} \)
79 \( 1 + 2.78iT - 79T^{2} \)
83 \( 1 + (-4.47 - 4.47i)T + 83iT^{2} \)
89 \( 1 + 2.53T + 89T^{2} \)
97 \( 1 + (-13.1 + 13.1i)T - 97iT^{2} \)
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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

−9.831443127002425674252632658383, −9.393687981360317351109853395255, −7.86487444260039429888139449808, −7.34994290725387290898938811634, −6.12082574649046268854566803853, −5.52331194618721623791241562146, −5.00215210736041845190884100417, −3.67381362461959444400703516318, −2.39834258928324927195875864872, −0.70715188167630263986845253357, 1.96722358424577069006215880188, 3.03041387005845550898206361720, 3.45404103123539943053423659577, 5.00302447938121478530874069102, 5.70989500641085581523724451893, 6.72586074604712931280936897292, 7.69148236631030926743887513672, 8.461261325773564456101285224808, 9.742850265222915144219675327454, 10.39082439022921434410910218213

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