Properties

Label 2-115-23.22-c6-0-6
Degree $2$
Conductor $115$
Sign $-0.534 - 0.845i$
Analytic cond. $26.4562$
Root an. cond. $5.14356$
Motivic weight $6$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.26·2-s + 21.3·3-s − 58.8·4-s − 55.9i·5-s + 48.2·6-s − 378. i·7-s − 278.·8-s − 275.·9-s − 126. i·10-s + 1.68e3i·11-s − 1.25e3·12-s + 623.·13-s − 857. i·14-s − 1.19e3i·15-s + 3.13e3·16-s + 5.99e3i·17-s + ⋯
L(s)  = 1  + 0.283·2-s + 0.788·3-s − 0.919·4-s − 0.447i·5-s + 0.223·6-s − 1.10i·7-s − 0.544·8-s − 0.377·9-s − 0.126i·10-s + 1.26i·11-s − 0.725·12-s + 0.283·13-s − 0.312i·14-s − 0.352i·15-s + 0.765·16-s + 1.22i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.534 - 0.845i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.534 - 0.845i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(115\)    =    \(5 \cdot 23\)
Sign: $-0.534 - 0.845i$
Analytic conductor: \(26.4562\)
Root analytic conductor: \(5.14356\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{115} (91, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 115,\ (\ :3),\ -0.534 - 0.845i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(0.7811093066\)
\(L(\frac12)\) \(\approx\) \(0.7811093066\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + 55.9iT \)
23 \( 1 + (1.02e4 - 6.49e3i)T \)
good2 \( 1 - 2.26T + 64T^{2} \)
3 \( 1 - 21.3T + 729T^{2} \)
7 \( 1 + 378. iT - 1.17e5T^{2} \)
11 \( 1 - 1.68e3iT - 1.77e6T^{2} \)
13 \( 1 - 623.T + 4.82e6T^{2} \)
17 \( 1 - 5.99e3iT - 2.41e7T^{2} \)
19 \( 1 - 6.41e3iT - 4.70e7T^{2} \)
29 \( 1 + 3.03e4T + 5.94e8T^{2} \)
31 \( 1 - 3.07e4T + 8.87e8T^{2} \)
37 \( 1 + 9.81e3iT - 2.56e9T^{2} \)
41 \( 1 + 8.15e4T + 4.75e9T^{2} \)
43 \( 1 - 1.70e3iT - 6.32e9T^{2} \)
47 \( 1 - 1.48e4T + 1.07e10T^{2} \)
53 \( 1 - 1.13e5iT - 2.21e10T^{2} \)
59 \( 1 + 3.70e5T + 4.21e10T^{2} \)
61 \( 1 - 5.08e4iT - 5.15e10T^{2} \)
67 \( 1 - 4.59e5iT - 9.04e10T^{2} \)
71 \( 1 - 1.94e5T + 1.28e11T^{2} \)
73 \( 1 - 7.09e4T + 1.51e11T^{2} \)
79 \( 1 + 3.91e5iT - 2.43e11T^{2} \)
83 \( 1 - 1.64e5iT - 3.26e11T^{2} \)
89 \( 1 + 1.24e6iT - 4.96e11T^{2} \)
97 \( 1 + 1.91e4iT - 8.32e11T^{2} \)
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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

−13.03274950112739198667797719286, −12.05026472156203106711442529512, −10.36064644320983801693336145089, −9.540522882216686174503805677110, −8.422315918082383125684923180441, −7.61436321671497337344554014507, −5.84281431828515097605795949827, −4.35973978553116586540202904089, −3.61931582167600973226689155189, −1.60638399011293318979136235312, 0.21058413389048924018914985610, 2.59360544431521758254269232158, 3.47267258541211532203326984744, 5.15269121070493093103796138553, 6.20057665216183099665704588752, 8.064051434601405820220077386576, 8.823990221121001227408746890865, 9.539634267807092655095308877313, 11.16942566384929811746036592300, 12.11330708975047183111259323122

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