Properties

Label 2-315-7.6-c2-0-16
Degree $2$
Conductor $315$
Sign $0.283 + 0.959i$
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  + 0.112·2-s − 3.98·4-s + 2.23i·5-s + (−6.71 + 1.98i)7-s − 0.902·8-s + 0.252i·10-s + 15.8·11-s − 13.3i·13-s + (−0.758 + 0.224i)14-s + 15.8·16-s − 15.6i·17-s − 30.8i·19-s − 8.91i·20-s + 1.79·22-s − 3.63·23-s + ⋯
L(s)  = 1  + 0.0564·2-s − 0.996·4-s + 0.447i·5-s + (−0.959 + 0.283i)7-s − 0.112·8-s + 0.0252i·10-s + 1.44·11-s − 1.02i·13-s + (−0.0541 + 0.0160i)14-s + 0.990·16-s − 0.922i·17-s − 1.62i·19-s − 0.445i·20-s + 0.0814·22-s − 0.158·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.283 + 0.959i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.283 + 0.959i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $0.283 + 0.959i$
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: \((2,\ 315,\ (\ :1),\ 0.283 + 0.959i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.761039 - 0.568680i\)
\(L(\frac12)\) \(\approx\) \(0.761039 - 0.568680i\)
\(L(2)\) 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 \( 1 + (6.71 - 1.98i)T \)
good2 \( 1 - 0.112T + 4T^{2} \)
11 \( 1 - 15.8T + 121T^{2} \)
13 \( 1 + 13.3iT - 169T^{2} \)
17 \( 1 + 15.6iT - 289T^{2} \)
19 \( 1 + 30.8iT - 361T^{2} \)
23 \( 1 + 3.63T + 529T^{2} \)
29 \( 1 + 14.5T + 841T^{2} \)
31 \( 1 + 11.3iT - 961T^{2} \)
37 \( 1 - 17.3T + 1.36e3T^{2} \)
41 \( 1 - 27.9iT - 1.68e3T^{2} \)
43 \( 1 + 12.1T + 1.84e3T^{2} \)
47 \( 1 + 80.5iT - 2.20e3T^{2} \)
53 \( 1 - 55.9T + 2.80e3T^{2} \)
59 \( 1 - 79.5iT - 3.48e3T^{2} \)
61 \( 1 + 94.5iT - 3.72e3T^{2} \)
67 \( 1 + 103.T + 4.48e3T^{2} \)
71 \( 1 - 113.T + 5.04e3T^{2} \)
73 \( 1 - 20.3iT - 5.32e3T^{2} \)
79 \( 1 + 1.27T + 6.24e3T^{2} \)
83 \( 1 - 19.7iT - 6.88e3T^{2} \)
89 \( 1 + 131. iT - 7.92e3T^{2} \)
97 \( 1 + 12.4iT - 9.40e3T^{2} \)
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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

−11.33339544036957089333177191351, −10.06263225666400372065779383921, −9.379644602632509144541940903094, −8.688686957970767696185127282649, −7.29783245373434453423333085355, −6.33368791608110957667785212820, −5.21740765021331735542771485403, −3.93031835552245905174059968467, −2.89731836593158168678175697352, −0.51351499673355526191030925639, 1.37180657378001640621487188606, 3.73600700664571609948948454714, 4.18822987145833187804252853177, 5.75680506752339121267743478565, 6.60458085656122418736898002286, 7.995342698267772318061405468576, 9.061359914642503499656006926949, 9.515997204100666850263677664300, 10.49422563069097466310350430381, 11.95527208812897516524739825990

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