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Properties

Label 2-115-5.4-c3-0-8
Degree $2$
Conductor $115$
Sign $0.253 - 0.967i$
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  − 0.231i·2-s + 2.18i·3-s + 7.94·4-s + (−10.8 − 2.83i)5-s + 0.506·6-s + 29.8i·7-s − 3.68i·8-s + 22.2·9-s + (−0.656 + 2.50i)10-s − 17.9·11-s + 17.3i·12-s + 35.4i·13-s + 6.90·14-s + (6.21 − 23.6i)15-s + 62.7·16-s + 68.5i·17-s + ⋯
L(s)  = 1  − 0.0817i·2-s + 0.421i·3-s + 0.993·4-s + (−0.967 − 0.253i)5-s + 0.0344·6-s + 1.61i·7-s − 0.163i·8-s + 0.822·9-s + (−0.0207 + 0.0791i)10-s − 0.493·11-s + 0.418i·12-s + 0.757i·13-s + 0.131·14-s + (0.106 − 0.407i)15-s + 0.979·16-s + 0.977i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.253 - 0.967i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.253 - 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(115\)    =    \(5 \cdot 23\)
Sign: $0.253 - 0.967i$
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: \((2,\ 115,\ (\ :3/2),\ 0.253 - 0.967i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.30447 + 1.00616i\)
\(L(\frac12)\) \(\approx\) \(1.30447 + 1.00616i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (10.8 + 2.83i)T \)
23 \( 1 - 23iT \)
good2 \( 1 + 0.231iT - 8T^{2} \)
3 \( 1 - 2.18iT - 27T^{2} \)
7 \( 1 - 29.8iT - 343T^{2} \)
11 \( 1 + 17.9T + 1.33e3T^{2} \)
13 \( 1 - 35.4iT - 2.19e3T^{2} \)
17 \( 1 - 68.5iT - 4.91e3T^{2} \)
19 \( 1 + 3.31T + 6.85e3T^{2} \)
29 \( 1 + 96.3T + 2.43e4T^{2} \)
31 \( 1 - 191.T + 2.97e4T^{2} \)
37 \( 1 + 250. iT - 5.06e4T^{2} \)
41 \( 1 + 96.7T + 6.89e4T^{2} \)
43 \( 1 + 84.4iT - 7.95e4T^{2} \)
47 \( 1 + 587. iT - 1.03e5T^{2} \)
53 \( 1 + 429. iT - 1.48e5T^{2} \)
59 \( 1 - 587.T + 2.05e5T^{2} \)
61 \( 1 + 613.T + 2.26e5T^{2} \)
67 \( 1 + 800. iT - 3.00e5T^{2} \)
71 \( 1 + 79.4T + 3.57e5T^{2} \)
73 \( 1 - 957. iT - 3.89e5T^{2} \)
79 \( 1 - 938.T + 4.93e5T^{2} \)
83 \( 1 - 396. iT - 5.71e5T^{2} \)
89 \( 1 - 292.T + 7.04e5T^{2} \)
97 \( 1 + 82.7iT - 9.12e5T^{2} \)
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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

−12.89111644033379711603307588791, −12.12483609237257404634061067639, −11.40133090471595965538455417127, −10.28322057404462339521243610422, −8.974624865127929341323439036324, −7.903301435710210375464350704065, −6.64034377894702057020461803961, −5.26170110702557484454822815545, −3.70287503079473226549507936035, −2.06452275723815072663698186456, 0.924687049679609649581125187461, 3.06686392232789001249701039963, 4.51638984050042170747337094929, 6.55546606401302711790552746722, 7.45682104835931750630643311405, 7.85315208305684649390129761606, 10.11624187276338351189854651829, 10.77961434270386180714868642761, 11.77683970373207363592639684577, 12.83089088914190980059867960337

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