Properties

Label 2-115-115.114-c2-0-20
Degree $2$
Conductor $115$
Sign $-0.593 - 0.804i$
Analytic cond. $3.13352$
Root an. cond. $1.77017$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3.64i·2-s − 3.06i·3-s − 9.32·4-s + (3.24 + 3.80i)5-s − 11.1·6-s − 12.1·7-s + 19.4i·8-s − 0.397·9-s + (13.8 − 11.8i)10-s − 10.9i·11-s + 28.5i·12-s − 17.9i·13-s + 44.4i·14-s + (11.6 − 9.94i)15-s + 33.5·16-s − 0.374·17-s + ⋯
L(s)  = 1  − 1.82i·2-s − 1.02i·3-s − 2.33·4-s + (0.648 + 0.761i)5-s − 1.86·6-s − 1.74·7-s + 2.42i·8-s − 0.0441·9-s + (1.38 − 1.18i)10-s − 0.991i·11-s + 2.38i·12-s − 1.38i·13-s + 3.17i·14-s + (0.777 − 0.662i)15-s + 2.09·16-s − 0.0220·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.593 - 0.804i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.593 - 0.804i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(115\)    =    \(5 \cdot 23\)
Sign: $-0.593 - 0.804i$
Analytic conductor: \(3.13352\)
Root analytic conductor: \(1.77017\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{115} (114, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 115,\ (\ :1),\ -0.593 - 0.804i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.429241 + 0.850066i\)
\(L(\frac12)\) \(\approx\) \(0.429241 + 0.850066i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-3.24 - 3.80i)T \)
23 \( 1 + (-22.9 - 1.61i)T \)
good2 \( 1 + 3.64iT - 4T^{2} \)
3 \( 1 + 3.06iT - 9T^{2} \)
7 \( 1 + 12.1T + 49T^{2} \)
11 \( 1 + 10.9iT - 121T^{2} \)
13 \( 1 + 17.9iT - 169T^{2} \)
17 \( 1 + 0.374T + 289T^{2} \)
19 \( 1 + 10.3iT - 361T^{2} \)
29 \( 1 + 11.7T + 841T^{2} \)
31 \( 1 - 32.5T + 961T^{2} \)
37 \( 1 + 25.4T + 1.36e3T^{2} \)
41 \( 1 - 13.8T + 1.68e3T^{2} \)
43 \( 1 + 14.5T + 1.84e3T^{2} \)
47 \( 1 + 41.7iT - 2.20e3T^{2} \)
53 \( 1 - 68.2T + 2.80e3T^{2} \)
59 \( 1 - 9.15T + 3.48e3T^{2} \)
61 \( 1 - 63.7iT - 3.72e3T^{2} \)
67 \( 1 + 4.06T + 4.48e3T^{2} \)
71 \( 1 - 47.2T + 5.04e3T^{2} \)
73 \( 1 - 80.6iT - 5.32e3T^{2} \)
79 \( 1 + 74.1iT - 6.24e3T^{2} \)
83 \( 1 + 151.T + 6.88e3T^{2} \)
89 \( 1 + 84.2iT - 7.92e3T^{2} \)
97 \( 1 - 36.3T + 9.40e3T^{2} \)
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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

−12.92882497098118934085980614910, −11.68703218579675178701205385161, −10.50190332500146369860428549604, −9.936520806547463002437492856826, −8.754769317124465174845486279011, −6.99794670215348213811343666293, −5.76641137889373214394132520603, −3.36397873015322146361638023707, −2.63742707375629511775891078153, −0.70485666729692346157638270894, 4.04420059239499078921151159765, 4.99923750125030875157266259776, 6.27214740505937126935858238144, 7.08758794407032861849276040128, 8.863385186021210181211928146083, 9.502646796977463685316771961131, 10.02538188996166681031290400651, 12.48698858813862956920489404872, 13.27246597319189333911081261129, 14.25040835811141462041220067485

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