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Properties

Label 2-115-23.22-c6-0-41
Degree $2$
Conductor $115$
Sign $-0.595 + 0.803i$
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  + 12.0·2-s − 19.2·3-s + 81.5·4-s + 55.9i·5-s − 232.·6-s − 145. i·7-s + 211.·8-s − 358.·9-s + 674. i·10-s − 1.53e3i·11-s − 1.56e3·12-s + 313.·13-s − 1.74e3i·14-s − 1.07e3i·15-s − 2.66e3·16-s − 4.25e3i·17-s + ⋯
L(s)  = 1  + 1.50·2-s − 0.712·3-s + 1.27·4-s + 0.447i·5-s − 1.07·6-s − 0.422i·7-s + 0.412·8-s − 0.492·9-s + 0.674i·10-s − 1.15i·11-s − 0.907·12-s + 0.142·13-s − 0.637i·14-s − 0.318i·15-s − 0.651·16-s − 0.865i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.595 + 0.803i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.595 + 0.803i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(115\)    =    \(5 \cdot 23\)
Sign: $-0.595 + 0.803i$
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: \((2,\ 115,\ (\ :3),\ -0.595 + 0.803i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(1.615941998\)
\(L(\frac12)\) \(\approx\) \(1.615941998\)
\(L(4)\) 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 \( 1 + (9.77e3 + 7.24e3i)T \)
good2 \( 1 - 12.0T + 64T^{2} \)
3 \( 1 + 19.2T + 729T^{2} \)
7 \( 1 + 145. iT - 1.17e5T^{2} \)
11 \( 1 + 1.53e3iT - 1.77e6T^{2} \)
13 \( 1 - 313.T + 4.82e6T^{2} \)
17 \( 1 + 4.25e3iT - 2.41e7T^{2} \)
19 \( 1 + 2.86e3iT - 4.70e7T^{2} \)
29 \( 1 + 7.85e3T + 5.94e8T^{2} \)
31 \( 1 + 1.03e4T + 8.87e8T^{2} \)
37 \( 1 + 3.42e4iT - 2.56e9T^{2} \)
41 \( 1 + 1.10e5T + 4.75e9T^{2} \)
43 \( 1 + 4.72e4iT - 6.32e9T^{2} \)
47 \( 1 + 4.91e4T + 1.07e10T^{2} \)
53 \( 1 - 1.36e5iT - 2.21e10T^{2} \)
59 \( 1 - 3.64e5T + 4.21e10T^{2} \)
61 \( 1 - 2.24e5iT - 5.15e10T^{2} \)
67 \( 1 - 5.00e5iT - 9.04e10T^{2} \)
71 \( 1 - 5.26e4T + 1.28e11T^{2} \)
73 \( 1 - 1.52e5T + 1.51e11T^{2} \)
79 \( 1 + 6.85e5iT - 2.43e11T^{2} \)
83 \( 1 + 3.79e5iT - 3.26e11T^{2} \)
89 \( 1 + 5.98e4iT - 4.96e11T^{2} \)
97 \( 1 - 6.60e5iT - 8.32e11T^{2} \)
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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

−11.96830571535328175688122611692, −11.39721824280461264561748931254, −10.49112256905009473113711256793, −8.731799494527511715185934126837, −7.03675064775465097613168885693, −6.03644226891495916097350382243, −5.22835059062429542015627843774, −3.85252833497382842763882911915, −2.70711543715005285551402322812, −0.32143790115590769039764922485, 1.97416917270447127272301169531, 3.65910776652748366150233125652, 4.90225657211520292554504337755, 5.68354498873830072906278638571, 6.67068946279553494121251423538, 8.331354633849423093199008317075, 9.813567188789604937388615959369, 11.22583709611307028502405561274, 12.09150613462488934720836023312, 12.61726463938566777464534275688

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