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Properties

Label 2-315-7.6-c2-0-24
Degree $2$
Conductor $315$
Sign $0.804 + 0.593i$
Analytic cond. $8.58312$
Root an. cond. $2.92969$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.79·2-s + 3.79·4-s − 2.23i·5-s + (4.15 − 5.63i)7-s − 0.578·8-s − 6.24i·10-s + 18.9·11-s − 10.9i·13-s + (11.6 − 15.7i)14-s − 16.7·16-s + 22.3i·17-s − 19.6i·19-s − 8.48i·20-s + 52.9·22-s + 31.9·23-s + ⋯
L(s)  = 1  + 1.39·2-s + 0.948·4-s − 0.447i·5-s + (0.593 − 0.804i)7-s − 0.0723·8-s − 0.624i·10-s + 1.72·11-s − 0.844i·13-s + (0.829 − 1.12i)14-s − 1.04·16-s + 1.31i·17-s − 1.03i·19-s − 0.424i·20-s + 2.40·22-s + 1.39·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.804 + 0.593i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.804 + 0.593i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $0.804 + 0.593i$
Analytic conductor: \(8.58312\)
Root analytic conductor: \(2.92969\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (181, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :1),\ 0.804 + 0.593i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(3.38073 - 1.11282i\)
\(L(\frac12)\) \(\approx\) \(3.38073 - 1.11282i\)
\(L(2)\) 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 \( 1 + 2.23iT \)
7 \( 1 + (-4.15 + 5.63i)T \)
good2 \( 1 - 2.79T + 4T^{2} \)
11 \( 1 - 18.9T + 121T^{2} \)
13 \( 1 + 10.9iT - 169T^{2} \)
17 \( 1 - 22.3iT - 289T^{2} \)
19 \( 1 + 19.6iT - 361T^{2} \)
23 \( 1 - 31.9T + 529T^{2} \)
29 \( 1 + 39.9T + 841T^{2} \)
31 \( 1 - 36.6iT - 961T^{2} \)
37 \( 1 - 8.94T + 1.36e3T^{2} \)
41 \( 1 - 37.6iT - 1.68e3T^{2} \)
43 \( 1 + 18.8T + 1.84e3T^{2} \)
47 \( 1 - 49.3iT - 2.20e3T^{2} \)
53 \( 1 + 49.2T + 2.80e3T^{2} \)
59 \( 1 + 35.2iT - 3.48e3T^{2} \)
61 \( 1 - 63.4iT - 3.72e3T^{2} \)
67 \( 1 - 21.3T + 4.48e3T^{2} \)
71 \( 1 + 36.2T + 5.04e3T^{2} \)
73 \( 1 + 6.66iT - 5.32e3T^{2} \)
79 \( 1 + 16.2T + 6.24e3T^{2} \)
83 \( 1 - 36.7iT - 6.88e3T^{2} \)
89 \( 1 - 88.0iT - 7.92e3T^{2} \)
97 \( 1 - 133. iT - 9.40e3T^{2} \)
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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

−11.46137345572885408883638419086, −10.88963995350645020594535212072, −9.413764901798267266469709425241, −8.509548967037549689460287965234, −7.15032983952679100751202064940, −6.23719053299752423885667334140, −5.07123077756916848592027182748, −4.24255848994682733147031803463, −3.32271323331046301987260367674, −1.33488933552444896048154572834, 1.98571923767132705310233485934, 3.39445959685017575912236551493, 4.38424598735040362730196818117, 5.45907339821238527069125915238, 6.39162211110912856972510314723, 7.28810948517769941856120706568, 8.914923304667216938300974729347, 9.482163503646378069021948099679, 11.33560004737655926432968634759, 11.57624871546075494794824925756

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