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Properties

Label 2-945-945.664-c0-0-1
Degree $2$
Conductor $945$
Sign $0.957 - 0.286i$
Analytic cond. $0.471616$
Root an. cond. $0.686743$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.5 − 0.866i)3-s + (0.173 + 0.984i)4-s + (0.939 + 0.342i)5-s + (−0.173 + 0.984i)7-s + (−0.499 − 0.866i)9-s + (−1.43 + 0.524i)11-s + (0.939 + 0.342i)12-s + (1.43 − 1.20i)13-s + (0.766 − 0.642i)15-s + (−0.939 + 0.342i)16-s + (0.173 + 0.300i)17-s + (−0.173 + 0.984i)20-s + (0.766 + 0.642i)21-s + (0.766 + 0.642i)25-s − 0.999·27-s − 0.999·28-s + ⋯
L(s)  = 1  + (0.5 − 0.866i)3-s + (0.173 + 0.984i)4-s + (0.939 + 0.342i)5-s + (−0.173 + 0.984i)7-s + (−0.499 − 0.866i)9-s + (−1.43 + 0.524i)11-s + (0.939 + 0.342i)12-s + (1.43 − 1.20i)13-s + (0.766 − 0.642i)15-s + (−0.939 + 0.342i)16-s + (0.173 + 0.300i)17-s + (−0.173 + 0.984i)20-s + (0.766 + 0.642i)21-s + (0.766 + 0.642i)25-s − 0.999·27-s − 0.999·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 945 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.957 - 0.286i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 945 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.957 - 0.286i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(945\)    =    \(3^{3} \cdot 5 \cdot 7\)
Sign: $0.957 - 0.286i$
Analytic conductor: \(0.471616\)
Root analytic conductor: \(0.686743\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{945} (664, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 945,\ (\ :0),\ 0.957 - 0.286i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.324590531\)
\(L(\frac12)\) \(\approx\) \(1.324590531\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.5 + 0.866i)T \)
5 \( 1 + (-0.939 - 0.342i)T \)
7 \( 1 + (0.173 - 0.984i)T \)
good2 \( 1 + (-0.173 - 0.984i)T^{2} \)
11 \( 1 + (1.43 - 0.524i)T + (0.766 - 0.642i)T^{2} \)
13 \( 1 + (-1.43 + 1.20i)T + (0.173 - 0.984i)T^{2} \)
17 \( 1 + (-0.173 - 0.300i)T + (-0.5 + 0.866i)T^{2} \)
19 \( 1 + (0.5 + 0.866i)T^{2} \)
23 \( 1 + (0.939 - 0.342i)T^{2} \)
29 \( 1 + (0.766 + 0.642i)T + (0.173 + 0.984i)T^{2} \)
31 \( 1 + (0.939 - 0.342i)T^{2} \)
37 \( 1 + (0.5 - 0.866i)T^{2} \)
41 \( 1 + (-0.173 + 0.984i)T^{2} \)
43 \( 1 + (-0.766 + 0.642i)T^{2} \)
47 \( 1 + (-0.326 + 1.85i)T + (-0.939 - 0.342i)T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + (-0.766 - 0.642i)T^{2} \)
61 \( 1 + (0.939 + 0.342i)T^{2} \)
67 \( 1 + (-0.173 + 0.984i)T^{2} \)
71 \( 1 + (0.173 + 0.300i)T + (-0.5 + 0.866i)T^{2} \)
73 \( 1 + (-0.173 + 0.300i)T + (-0.5 - 0.866i)T^{2} \)
79 \( 1 + (-1.17 - 0.984i)T + (0.173 + 0.984i)T^{2} \)
83 \( 1 + (1.17 + 0.984i)T + (0.173 + 0.984i)T^{2} \)
89 \( 1 + (0.5 + 0.866i)T^{2} \)
97 \( 1 + (-0.326 + 0.118i)T + (0.766 - 0.642i)T^{2} \)
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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

−10.27673240953728247523478905489, −9.242185500372210266113700593480, −8.399856853141818940112802374738, −7.938673844541462270220393500236, −6.98242773992406875013165437071, −6.02518434818359465430697568868, −5.40782305866224790548944229005, −3.52573713133103679156382105390, −2.74700512781311644572523814530, −2.00131690612319785266331684650, 1.47726320217678394754248188102, 2.76033540547715234004015053837, 4.04394670215629859157687838483, 5.00827252477979832075528635483, 5.74711571541995755438577367335, 6.61074089667813627694415133217, 7.82347175607991525948861819101, 8.882523644664486181853183203947, 9.441064163990443444669390295660, 10.23011631473567440796612113991

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