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Properties

Label 2-945-315.79-c1-0-20
Degree $2$
Conductor $945$
Sign $0.138 + 0.990i$
Analytic cond. $7.54586$
Root an. cond. $2.74697$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.866 + 0.5i)2-s + (−0.500 + 0.866i)4-s + (−1 + 2i)5-s + (−0.866 + 2.5i)7-s − 3i·8-s + (−0.133 − 2.23i)10-s − 6·11-s + (3.46 − 2i)13-s + (−0.500 − 2.59i)14-s + (0.500 + 0.866i)16-s + (−1.73 + i)17-s + (−3 + 5.19i)19-s + (−1.23 − 1.86i)20-s + (5.19 − 3i)22-s − 3i·23-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (−0.250 + 0.433i)4-s + (−0.447 + 0.894i)5-s + (−0.327 + 0.944i)7-s − 1.06i·8-s + (−0.0423 − 0.705i)10-s − 1.80·11-s + (0.960 − 0.554i)13-s + (−0.133 − 0.694i)14-s + (0.125 + 0.216i)16-s + (−0.420 + 0.242i)17-s + (−0.688 + 1.19i)19-s + (−0.275 − 0.417i)20-s + (1.10 − 0.639i)22-s − 0.625i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 945 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.138 + 0.990i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 945 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.138 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(945\)    =    \(3^{3} \cdot 5 \cdot 7\)
Sign: $0.138 + 0.990i$
Analytic conductor: \(7.54586\)
Root analytic conductor: \(2.74697\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{945} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 945,\ (\ :1/2),\ 0.138 + 0.990i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + (1 - 2i)T \)
7 \( 1 + (0.866 - 2.5i)T \)
good2 \( 1 + (0.866 - 0.5i)T + (1 - 1.73i)T^{2} \)
11 \( 1 + 6T + 11T^{2} \)
13 \( 1 + (-3.46 + 2i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (1.73 - i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (3 - 5.19i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + 3iT - 23T^{2} \)
29 \( 1 + (-1 + 1.73i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-2 + 3.46i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-6.92 - 4i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-1 - 1.73i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.866 - 0.5i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (2.59 - 1.5i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-10.3 + 6i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (2 - 3.46i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (3.5 + 6.06i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (6.06 + 3.5i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 4T + 71T^{2} \)
73 \( 1 + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (7 + 12.1i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (3.46 + 2i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (-1.5 + 2.59i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (1.73 + i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.983651266167251954008226532773, −8.724813432444725111887427415178, −8.122019168472741418224421908468, −7.68392073890605511931703098410, −6.43600042673380661162013142941, −5.86252284168097154177179963555, −4.41911360431931481173671015278, −3.29898561998205251261257790986, −2.49409379912129157296858947865, 0, 1.13166165923951235891746322642, 2.61186592818155495333887128264, 4.13565110848206043347594163870, 4.87822260967141270789919566581, 5.78266340669509851318036502795, 7.06712517934449281138050480897, 7.957891718573052861656574671983, 8.709093203300234142693482288040, 9.335057153538289899389028323759

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