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Properties

Label 2-115-23.22-c6-0-18
Degree $2$
Conductor $115$
Sign $0.482 - 0.875i$
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  − 4.89·2-s + 32.2·3-s − 40.0·4-s + 55.9i·5-s − 157.·6-s + 6.78i·7-s + 509.·8-s + 310.·9-s − 273. i·10-s − 1.15e3i·11-s − 1.29e3·12-s + 1.14e3·13-s − 33.1i·14-s + 1.80e3i·15-s + 75.1·16-s + 5.26e3i·17-s + ⋯
L(s)  = 1  − 0.611·2-s + 1.19·3-s − 0.626·4-s + 0.447i·5-s − 0.730·6-s + 0.0197i·7-s + 0.994·8-s + 0.426·9-s − 0.273i·10-s − 0.870i·11-s − 0.747·12-s + 0.519·13-s − 0.0120i·14-s + 0.534i·15-s + 0.0183·16-s + 1.07i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.482 - 0.875i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.482 - 0.875i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(115\)    =    \(5 \cdot 23\)
Sign: $0.482 - 0.875i$
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: \((2,\ 115,\ (\ :3),\ 0.482 - 0.875i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(1.659017110\)
\(L(\frac12)\) \(\approx\) \(1.659017110\)
\(L(4)\) 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 \( 1 + (-1.06e4 - 5.87e3i)T \)
good2 \( 1 + 4.89T + 64T^{2} \)
3 \( 1 - 32.2T + 729T^{2} \)
7 \( 1 - 6.78iT - 1.17e5T^{2} \)
11 \( 1 + 1.15e3iT - 1.77e6T^{2} \)
13 \( 1 - 1.14e3T + 4.82e6T^{2} \)
17 \( 1 - 5.26e3iT - 2.41e7T^{2} \)
19 \( 1 - 2.82e3iT - 4.70e7T^{2} \)
29 \( 1 + 8.19e3T + 5.94e8T^{2} \)
31 \( 1 - 1.94e4T + 8.87e8T^{2} \)
37 \( 1 - 6.70e4iT - 2.56e9T^{2} \)
41 \( 1 - 1.14e5T + 4.75e9T^{2} \)
43 \( 1 - 1.00e5iT - 6.32e9T^{2} \)
47 \( 1 + 1.65e4T + 1.07e10T^{2} \)
53 \( 1 - 2.23e5iT - 2.21e10T^{2} \)
59 \( 1 - 2.95e4T + 4.21e10T^{2} \)
61 \( 1 - 1.57e5iT - 5.15e10T^{2} \)
67 \( 1 - 4.21e5iT - 9.04e10T^{2} \)
71 \( 1 + 1.81e5T + 1.28e11T^{2} \)
73 \( 1 - 5.58e5T + 1.51e11T^{2} \)
79 \( 1 + 2.68e5iT - 2.43e11T^{2} \)
83 \( 1 + 3.87e5iT - 3.26e11T^{2} \)
89 \( 1 + 6.67e5iT - 4.96e11T^{2} \)
97 \( 1 + 1.19e6iT - 8.32e11T^{2} \)
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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

−12.97699804432139473433592911664, −11.24519663717953868189577289649, −10.22602533159560122485854150437, −9.130078643903186123422903652334, −8.429856469756499229325219543962, −7.62525846156617169602278213110, −5.93207027801075606659726542008, −4.08149766743143645136089899023, −2.97309252194196549879064419476, −1.25411412993062537833984427952, 0.67554880806071703664915745621, 2.28530009729005571041783982617, 3.89630979635911219692819805106, 5.10387192276444713936591796030, 7.19324492882897427687584273943, 8.161781313691505128555711284745, 9.102318140623378901477452566783, 9.551220717388582369940633582037, 10.92345869352972941233839918188, 12.49390298034366454349896970743

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