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Properties

Label 2-945-945.524-c1-0-94
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  + (−1.62 + 0.586i)3-s + (−0.347 − 1.96i)4-s + (0.764 − 2.10i)5-s + (0.459 − 2.60i)7-s + (2.31 − 1.91i)9-s + (2.20 + 6.07i)11-s + (1.72 + 3.00i)12-s + (3.66 − 3.07i)13-s + (−0.0137 + 3.87i)15-s + (−3.75 + 1.36i)16-s + (6.43 − 3.71i)17-s + (−4.40 − 0.776i)20-s + (0.779 + 4.51i)21-s + (−3.83 − 3.21i)25-s + (−2.64 + 4.47i)27-s − 5.29·28-s + ⋯
L(s)  = 1  + (−0.940 + 0.338i)3-s + (−0.173 − 0.984i)4-s + (0.342 − 0.939i)5-s + (0.173 − 0.984i)7-s + (0.770 − 0.637i)9-s + (0.666 + 1.83i)11-s + (0.496 + 0.867i)12-s + (1.01 − 0.852i)13-s + (−0.00354 + 0.999i)15-s + (−0.939 + 0.342i)16-s + (1.56 − 0.900i)17-s + (−0.984 − 0.173i)20-s + (0.170 + 0.985i)21-s + (−0.766 − 0.642i)25-s + (−0.509 + 0.860i)27-s − 0.999·28-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: $\chi_{945} (524, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 945,\ (\ :1/2),\ -0.240 + 0.970i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.792616 - 1.01344i\)
\(L(\frac12)\) \(\approx\) \(0.792616 - 1.01344i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.62 - 0.586i)T \)
5 \( 1 + (-0.764 + 2.10i)T \)
7 \( 1 + (-0.459 + 2.60i)T \)
good2 \( 1 + (0.347 + 1.96i)T^{2} \)
11 \( 1 + (-2.20 - 6.07i)T + (-8.42 + 7.07i)T^{2} \)
13 \( 1 + (-3.66 + 3.07i)T + (2.25 - 12.8i)T^{2} \)
17 \( 1 + (-6.43 + 3.71i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-21.6 + 7.86i)T^{2} \)
29 \( 1 + (-3.10 + 3.70i)T + (-5.03 - 28.5i)T^{2} \)
31 \( 1 + (29.1 - 10.6i)T^{2} \)
37 \( 1 + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (7.11 - 40.3i)T^{2} \)
43 \( 1 + (-32.9 + 27.6i)T^{2} \)
47 \( 1 + (13.0 + 2.29i)T + (44.1 + 16.0i)T^{2} \)
53 \( 1 + 53T^{2} \)
59 \( 1 + (45.1 + 37.9i)T^{2} \)
61 \( 1 + (57.3 + 20.8i)T^{2} \)
67 \( 1 + (-11.6 + 65.9i)T^{2} \)
71 \( 1 + (-12.0 + 6.93i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (5.68 - 9.85i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-8.15 - 6.84i)T + (13.7 + 77.7i)T^{2} \)
83 \( 1 + (2.15 - 2.56i)T + (-14.4 - 81.7i)T^{2} \)
89 \( 1 + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-9.22 + 3.35i)T + (74.3 - 62.3i)T^{2} \)
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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

−9.808896203690418902630315127613, −9.524743576827686310243806295690, −8.117370613895400291432534289805, −7.05508850314297519789649286711, −6.21831470108974496948343475234, −5.25689209473588680289003903183, −4.75042439083546374059591646313, −3.87459944637849929639090829658, −1.52960159803756590908302495116, −0.802226445511832676240444895015, 1.53857621529139413314050375306, 3.08130532471539468907167993469, 3.79746021896425250992508826177, 5.31649736041447417562701703239, 6.21620712013033741507590106474, 6.54059551925184470221017071566, 7.85346285000633607397069949658, 8.479207935192030809987696098811, 9.370845352798764777637800279411, 10.56082589996792808651191936071

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