Properties

Label 2-115-115.114-c4-0-23
Degree $2$
Conductor $115$
Sign $0.880 + 0.473i$
Analytic cond. $11.8875$
Root an. cond. $3.44783$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.05i·2-s − 9.44i·3-s + 14.8·4-s + (−2.52 + 24.8i)5-s − 10.0·6-s + 67.6·7-s − 32.7i·8-s − 8.12·9-s + (26.3 + 2.67i)10-s + 168. i·11-s − 140. i·12-s + 214. i·13-s − 71.7i·14-s + (234. + 23.8i)15-s + 203.·16-s − 13.4·17-s + ⋯
L(s)  = 1  − 0.264i·2-s − 1.04i·3-s + 0.929·4-s + (−0.101 + 0.994i)5-s − 0.277·6-s + 1.38·7-s − 0.511i·8-s − 0.100·9-s + (0.263 + 0.0267i)10-s + 1.38i·11-s − 0.975i·12-s + 1.27i·13-s − 0.366i·14-s + (1.04 + 0.106i)15-s + 0.794·16-s − 0.0463·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.880 + 0.473i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.880 + 0.473i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(115\)    =    \(5 \cdot 23\)
Sign: $0.880 + 0.473i$
Analytic conductor: \(11.8875\)
Root analytic conductor: \(3.44783\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{115} (114, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 115,\ (\ :2),\ 0.880 + 0.473i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(2.43938 - 0.614539i\)
\(L(\frac12)\) \(\approx\) \(2.43938 - 0.614539i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (2.52 - 24.8i)T \)
23 \( 1 + (-202. + 488. i)T \)
good2 \( 1 + 1.05iT - 16T^{2} \)
3 \( 1 + 9.44iT - 81T^{2} \)
7 \( 1 - 67.6T + 2.40e3T^{2} \)
11 \( 1 - 168. iT - 1.46e4T^{2} \)
13 \( 1 - 214. iT - 2.85e4T^{2} \)
17 \( 1 + 13.4T + 8.35e4T^{2} \)
19 \( 1 + 362. iT - 1.30e5T^{2} \)
29 \( 1 + 1.21e3T + 7.07e5T^{2} \)
31 \( 1 - 1.39e3T + 9.23e5T^{2} \)
37 \( 1 - 1.06e3T + 1.87e6T^{2} \)
41 \( 1 + 541.T + 2.82e6T^{2} \)
43 \( 1 + 2.34e3T + 3.41e6T^{2} \)
47 \( 1 + 416. iT - 4.87e6T^{2} \)
53 \( 1 + 4.14e3T + 7.89e6T^{2} \)
59 \( 1 + 857.T + 1.21e7T^{2} \)
61 \( 1 - 3.04e3iT - 1.38e7T^{2} \)
67 \( 1 - 1.87e3T + 2.01e7T^{2} \)
71 \( 1 - 9.87e3T + 2.54e7T^{2} \)
73 \( 1 + 6.97e3iT - 2.83e7T^{2} \)
79 \( 1 - 2.45e3iT - 3.89e7T^{2} \)
83 \( 1 + 1.22e4T + 4.74e7T^{2} \)
89 \( 1 - 8.56e3iT - 6.27e7T^{2} \)
97 \( 1 + 8.34e3T + 8.85e7T^{2} \)
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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

−12.53413534658534197989560329839, −11.62035444018891115821290601418, −11.09331811608325465646965185137, −9.820890911895211761617658541483, −7.975250239775052126666462585819, −7.08546417659202416938922684441, −6.58651165962830825798898462697, −4.50622301904416835592357788570, −2.37766849972395423804695269991, −1.63445498780254574499334687613, 1.35402621710588938524822371876, 3.49150981616805223685491383356, 5.01533817897392517263168965137, 5.76987835805759309610500450082, 7.84289270078122439264555354135, 8.367603603728335396409001805579, 9.849174229922201427071730840406, 11.00325728075792682845810927613, 11.53914227777214197841171266298, 12.89147645510807822839615646509

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