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Properties

Label 2-115-23.22-c6-0-4
Degree $2$
Conductor $115$
Sign $-0.959 - 0.280i$
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  + 8.20·2-s + 14.1·3-s + 3.31·4-s + 55.9i·5-s + 115.·6-s − 163. i·7-s − 497.·8-s − 529.·9-s + 458. i·10-s + 843. i·11-s + 46.8·12-s − 2.29e3·13-s − 1.33e3i·14-s + 789. i·15-s − 4.29e3·16-s + 6.17e3i·17-s + ⋯
L(s)  = 1  + 1.02·2-s + 0.523·3-s + 0.0518·4-s + 0.447i·5-s + 0.536·6-s − 0.475i·7-s − 0.972·8-s − 0.726·9-s + 0.458i·10-s + 0.634i·11-s + 0.0271·12-s − 1.04·13-s − 0.487i·14-s + 0.234i·15-s − 1.04·16-s + 1.25i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.959 - 0.280i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.959 - 0.280i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(115\)    =    \(5 \cdot 23\)
Sign: $-0.959 - 0.280i$
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: \((2,\ 115,\ (\ :3),\ -0.959 - 0.280i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(0.7073868189\)
\(L(\frac12)\) \(\approx\) \(0.7073868189\)
\(L(4)\) 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 \( 1 + (-3.41e3 + 1.16e4i)T \)
good2 \( 1 - 8.20T + 64T^{2} \)
3 \( 1 - 14.1T + 729T^{2} \)
7 \( 1 + 163. iT - 1.17e5T^{2} \)
11 \( 1 - 843. iT - 1.77e6T^{2} \)
13 \( 1 + 2.29e3T + 4.82e6T^{2} \)
17 \( 1 - 6.17e3iT - 2.41e7T^{2} \)
19 \( 1 + 1.33e3iT - 4.70e7T^{2} \)
29 \( 1 + 1.10e4T + 5.94e8T^{2} \)
31 \( 1 + 3.56e4T + 8.87e8T^{2} \)
37 \( 1 - 4.29e4iT - 2.56e9T^{2} \)
41 \( 1 - 2.36e4T + 4.75e9T^{2} \)
43 \( 1 - 1.06e4iT - 6.32e9T^{2} \)
47 \( 1 + 2.21e4T + 1.07e10T^{2} \)
53 \( 1 + 1.15e4iT - 2.21e10T^{2} \)
59 \( 1 + 1.27e5T + 4.21e10T^{2} \)
61 \( 1 - 1.28e5iT - 5.15e10T^{2} \)
67 \( 1 + 1.83e5iT - 9.04e10T^{2} \)
71 \( 1 + 1.37e5T + 1.28e11T^{2} \)
73 \( 1 - 1.26e5T + 1.51e11T^{2} \)
79 \( 1 - 6.01e5iT - 2.43e11T^{2} \)
83 \( 1 + 5.17e5iT - 3.26e11T^{2} \)
89 \( 1 - 6.61e3iT - 4.96e11T^{2} \)
97 \( 1 - 4.63e4iT - 8.32e11T^{2} \)
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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

−12.99801857726179241859702509066, −12.19039074411989151063540296269, −10.93243711872532056126071276173, −9.711661430626464781498586674574, −8.570952021776205260689090947878, −7.26177635177561777802135022801, −5.97975966882501253123523826917, −4.67758267012225929791503995110, −3.53529710376021031043955806864, −2.35154250347468183467440323806, 0.14016502875775234249668502670, 2.48644638974960392789396381129, 3.56371081350098763331670232404, 5.06275916296882247030601767195, 5.77251194977555516437370884921, 7.51627564438401121188103166609, 8.896053388187676292749179768386, 9.427005063111520846856261669993, 11.34044891436713965525538955379, 12.13976889053673290286363102363

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