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Properties

Label 2-115-5.4-c3-0-4
Degree $2$
Conductor $115$
Sign $-0.775 - 0.630i$
Analytic cond. $6.78521$
Root an. cond. $2.60484$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3.17i·2-s − 4.30i·3-s − 2.10·4-s + (−7.05 + 8.67i)5-s + 13.6·6-s − 0.115i·7-s + 18.7i·8-s + 8.46·9-s + (−27.5 − 22.4i)10-s − 24.6·11-s + 9.08i·12-s + 81.9i·13-s + 0.367·14-s + (37.3 + 30.3i)15-s − 76.4·16-s + 68.7i·17-s + ⋯
L(s)  = 1  + 1.12i·2-s − 0.828i·3-s − 0.263·4-s + (−0.630 + 0.775i)5-s + 0.931·6-s − 0.00624i·7-s + 0.827i·8-s + 0.313·9-s + (−0.872 − 0.709i)10-s − 0.676·11-s + 0.218i·12-s + 1.74i·13-s + 0.00701·14-s + (0.642 + 0.522i)15-s − 1.19·16-s + 0.980i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.775 - 0.630i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.775 - 0.630i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(115\)    =    \(5 \cdot 23\)
Sign: $-0.775 - 0.630i$
Analytic conductor: \(6.78521\)
Root analytic conductor: \(2.60484\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{115} (24, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 115,\ (\ :3/2),\ -0.775 - 0.630i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.433800 + 1.22144i\)
\(L(\frac12)\) \(\approx\) \(0.433800 + 1.22144i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (7.05 - 8.67i)T \)
23 \( 1 - 23iT \)
good2 \( 1 - 3.17iT - 8T^{2} \)
3 \( 1 + 4.30iT - 27T^{2} \)
7 \( 1 + 0.115iT - 343T^{2} \)
11 \( 1 + 24.6T + 1.33e3T^{2} \)
13 \( 1 - 81.9iT - 2.19e3T^{2} \)
17 \( 1 - 68.7iT - 4.91e3T^{2} \)
19 \( 1 + 44.0T + 6.85e3T^{2} \)
29 \( 1 - 266.T + 2.43e4T^{2} \)
31 \( 1 + 185.T + 2.97e4T^{2} \)
37 \( 1 - 44.7iT - 5.06e4T^{2} \)
41 \( 1 - 283.T + 6.89e4T^{2} \)
43 \( 1 + 351. iT - 7.95e4T^{2} \)
47 \( 1 - 390. iT - 1.03e5T^{2} \)
53 \( 1 - 140. iT - 1.48e5T^{2} \)
59 \( 1 - 354.T + 2.05e5T^{2} \)
61 \( 1 - 607.T + 2.26e5T^{2} \)
67 \( 1 + 724. iT - 3.00e5T^{2} \)
71 \( 1 + 523.T + 3.57e5T^{2} \)
73 \( 1 + 402. iT - 3.89e5T^{2} \)
79 \( 1 + 1.11e3T + 4.93e5T^{2} \)
83 \( 1 - 440. iT - 5.71e5T^{2} \)
89 \( 1 - 348.T + 7.04e5T^{2} \)
97 \( 1 + 582. iT - 9.12e5T^{2} \)
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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

−13.82110283960469754133217286671, −12.52463745753641311617651298863, −11.52321707670451017913480109376, −10.46757017845698078992043494424, −8.686155061901960832220664684277, −7.64558758123299934180404503483, −6.92551129736781715914015156023, −6.15706585429407115259196978036, −4.31799867617259823544986546240, −2.17940981420911745012657157082, 0.69711257232208946856992268550, 2.87835322484809800350263222457, 4.14147904413679632772241224480, 5.26766056782371445385998665358, 7.35706546379577361995418836912, 8.650435464629146792274301074917, 9.918469055126090410569802386062, 10.53339945740323836179521931829, 11.53274044274922705760828129893, 12.67208049176022240191410510507

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