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Properties

Label 2-115-5.4-c5-0-37
Degree $2$
Conductor $115$
Sign $0.717 + 0.696i$
Analytic cond. $18.4441$
Root an. cond. $4.29466$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 7.77i·2-s − 16.5i·3-s − 28.3·4-s + (38.9 − 40.1i)5-s + 128.·6-s + 7.31i·7-s + 28.0i·8-s − 29.6·9-s + (311. + 302. i)10-s − 512.·11-s + 468. i·12-s − 603. i·13-s − 56.8·14-s + (−662. − 642. i)15-s − 1.12e3·16-s − 1.54e3i·17-s + ⋯
L(s)  = 1  + 1.37i·2-s − 1.05i·3-s − 0.887·4-s + (0.696 − 0.717i)5-s + 1.45·6-s + 0.0563i·7-s + 0.154i·8-s − 0.121·9-s + (0.986 + 0.956i)10-s − 1.27·11-s + 0.939i·12-s − 0.991i·13-s − 0.0774·14-s + (−0.760 − 0.737i)15-s − 1.10·16-s − 1.29i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.717 + 0.696i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.717 + 0.696i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(115\)    =    \(5 \cdot 23\)
Sign: $0.717 + 0.696i$
Analytic conductor: \(18.4441\)
Root analytic conductor: \(4.29466\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{115} (24, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 115,\ (\ :5/2),\ 0.717 + 0.696i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.705741156\)
\(L(\frac12)\) \(\approx\) \(1.705741156\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-38.9 + 40.1i)T \)
23 \( 1 + 529iT \)
good2 \( 1 - 7.77iT - 32T^{2} \)
3 \( 1 + 16.5iT - 243T^{2} \)
7 \( 1 - 7.31iT - 1.68e4T^{2} \)
11 \( 1 + 512.T + 1.61e5T^{2} \)
13 \( 1 + 603. iT - 3.71e5T^{2} \)
17 \( 1 + 1.54e3iT - 1.41e6T^{2} \)
19 \( 1 - 2.17e3T + 2.47e6T^{2} \)
29 \( 1 + 2.11e3T + 2.05e7T^{2} \)
31 \( 1 + 6.35e3T + 2.86e7T^{2} \)
37 \( 1 - 3.40e3iT - 6.93e7T^{2} \)
41 \( 1 - 8.00e3T + 1.15e8T^{2} \)
43 \( 1 + 1.46e4iT - 1.47e8T^{2} \)
47 \( 1 + 2.29e3iT - 2.29e8T^{2} \)
53 \( 1 + 1.97e4iT - 4.18e8T^{2} \)
59 \( 1 - 4.84e4T + 7.14e8T^{2} \)
61 \( 1 + 2.26e3T + 8.44e8T^{2} \)
67 \( 1 - 4.77e4iT - 1.35e9T^{2} \)
71 \( 1 + 5.94e4T + 1.80e9T^{2} \)
73 \( 1 - 1.20e4iT - 2.07e9T^{2} \)
79 \( 1 + 5.90e4T + 3.07e9T^{2} \)
83 \( 1 + 8.82e4iT - 3.93e9T^{2} \)
89 \( 1 - 6.14e4T + 5.58e9T^{2} \)
97 \( 1 - 5.59e4iT - 8.58e9T^{2} \)
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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

−13.05099578560108901374199263475, −11.78073773620009542231417440208, −10.11604233167702972217796810931, −8.829000039741333777524552807061, −7.71267382719154389276273552270, −7.16842135174696814372738731983, −5.69561263286208808499789390957, −5.16077727678530791163660610043, −2.40238050959137791906522817291, −0.62126218084556640989897666572, 1.74097573759559664947671759342, 3.05102650454746914291374371272, 4.14693566911099187620510295807, 5.59865992238751739569236332933, 7.27777959910509069597051553058, 9.210602687510484403495837239746, 9.906480025673495044340984154370, 10.67175325187585483955577495022, 11.26104878594114844976988970464, 12.67440381456444354851817980420

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