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Properties

Label 2-115-5.4-c3-0-26
Degree $2$
Conductor $115$
Sign $-0.842 + 0.538i$
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  − 2.51i·2-s − 6.03i·3-s + 1.66·4-s + (6.02 + 9.42i)5-s − 15.2·6-s − 24.0i·7-s − 24.3i·8-s − 9.45·9-s + (23.7 − 15.1i)10-s − 3.60·11-s − 10.0i·12-s − 18.9i·13-s − 60.6·14-s + (56.8 − 36.3i)15-s − 47.9·16-s + 97.4i·17-s + ⋯
L(s)  = 1  − 0.890i·2-s − 1.16i·3-s + 0.207·4-s + (0.538 + 0.842i)5-s − 1.03·6-s − 1.30i·7-s − 1.07i·8-s − 0.350·9-s + (0.750 − 0.479i)10-s − 0.0987·11-s − 0.241i·12-s − 0.403i·13-s − 1.15·14-s + (0.979 − 0.625i)15-s − 0.749·16-s + 1.39i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.842 + 0.538i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.842 + 0.538i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(115\)    =    \(5 \cdot 23\)
Sign: $-0.842 + 0.538i$
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: \((2,\ 115,\ (\ :3/2),\ -0.842 + 0.538i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.535709 - 1.83294i\)
\(L(\frac12)\) \(\approx\) \(0.535709 - 1.83294i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-6.02 - 9.42i)T \)
23 \( 1 - 23iT \)
good2 \( 1 + 2.51iT - 8T^{2} \)
3 \( 1 + 6.03iT - 27T^{2} \)
7 \( 1 + 24.0iT - 343T^{2} \)
11 \( 1 + 3.60T + 1.33e3T^{2} \)
13 \( 1 + 18.9iT - 2.19e3T^{2} \)
17 \( 1 - 97.4iT - 4.91e3T^{2} \)
19 \( 1 + 73.2T + 6.85e3T^{2} \)
29 \( 1 - 165.T + 2.43e4T^{2} \)
31 \( 1 - 156.T + 2.97e4T^{2} \)
37 \( 1 + 83.4iT - 5.06e4T^{2} \)
41 \( 1 + 86.1T + 6.89e4T^{2} \)
43 \( 1 - 439. iT - 7.95e4T^{2} \)
47 \( 1 + 573. iT - 1.03e5T^{2} \)
53 \( 1 - 357. iT - 1.48e5T^{2} \)
59 \( 1 - 726.T + 2.05e5T^{2} \)
61 \( 1 - 354.T + 2.26e5T^{2} \)
67 \( 1 + 182. iT - 3.00e5T^{2} \)
71 \( 1 - 1.01e3T + 3.57e5T^{2} \)
73 \( 1 - 806. iT - 3.89e5T^{2} \)
79 \( 1 + 582.T + 4.93e5T^{2} \)
83 \( 1 + 292. iT - 5.71e5T^{2} \)
89 \( 1 - 531.T + 7.04e5T^{2} \)
97 \( 1 + 380. iT - 9.12e5T^{2} \)
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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

−12.82639971294716238675444442212, −11.59939191867895880970606365671, −10.48716141033817311773060184981, −10.14744900597812128649615015150, −8.042266137902335656154537301776, −6.93388852467227579711526631073, −6.33838000157885500051588464802, −3.84704632681645222383511280432, −2.34839185218303331429128257850, −1.10575591321643456054848380446, 2.44120596267004943292860958091, 4.69919138089143140899408068699, 5.43623106688404620091223515147, 6.62018797120102280761775329153, 8.373584620288267365411302279801, 9.082183529039191670980661938688, 10.08503583132458949264020004144, 11.46138410972566733649640386689, 12.39354589493571935189660841595, 13.85074783488878944594317829041

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