Properties

Label 2-117-13.12-c3-0-11
Degree $2$
Conductor $117$
Sign $0.927 + 0.373i$
Analytic cond. $6.90322$
Root an. cond. $2.62739$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5.21i·2-s − 19.2·4-s − 5.83i·5-s − 31.3i·7-s − 58.6i·8-s + 30.4·10-s + 16.2i·11-s + (−43.4 − 17.5i)13-s + 163.·14-s + 152.·16-s − 54·17-s − 66.3i·19-s + 112. i·20-s − 84.9·22-s − 182.·23-s + ⋯
L(s)  = 1  + 1.84i·2-s − 2.40·4-s − 0.522i·5-s − 1.69i·7-s − 2.59i·8-s + 0.963·10-s + 0.446i·11-s + (−0.927 − 0.373i)13-s + 3.11·14-s + 2.37·16-s − 0.770·17-s − 0.801i·19-s + 1.25i·20-s − 0.823·22-s − 1.65·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.927 + 0.373i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.927 + 0.373i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(117\)    =    \(3^{2} \cdot 13\)
Sign: $0.927 + 0.373i$
Analytic conductor: \(6.90322\)
Root analytic conductor: \(2.62739\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{117} (64, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 117,\ (\ :3/2),\ 0.927 + 0.373i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.692905 - 0.134351i\)
\(L(\frac12)\) \(\approx\) \(0.692905 - 0.134351i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
13 \( 1 + (43.4 + 17.5i)T \)
good2 \( 1 - 5.21iT - 8T^{2} \)
5 \( 1 + 5.83iT - 125T^{2} \)
7 \( 1 + 31.3iT - 343T^{2} \)
11 \( 1 - 16.2iT - 1.33e3T^{2} \)
17 \( 1 + 54T + 4.91e3T^{2} \)
19 \( 1 + 66.3iT - 6.85e3T^{2} \)
23 \( 1 + 182.T + 1.21e4T^{2} \)
29 \( 1 - 164.T + 2.43e4T^{2} \)
31 \( 1 - 58.9iT - 2.97e4T^{2} \)
37 \( 1 + 110. iT - 5.06e4T^{2} \)
41 \( 1 - 55.0iT - 6.89e4T^{2} \)
43 \( 1 - 113.T + 7.95e4T^{2} \)
47 \( 1 + 514. iT - 1.03e5T^{2} \)
53 \( 1 + 242.T + 1.48e5T^{2} \)
59 \( 1 - 265. iT - 2.05e5T^{2} \)
61 \( 1 + 468.T + 2.26e5T^{2} \)
67 \( 1 + 852. iT - 3.00e5T^{2} \)
71 \( 1 - 165. iT - 3.57e5T^{2} \)
73 \( 1 - 315. iT - 3.89e5T^{2} \)
79 \( 1 - 479.T + 4.93e5T^{2} \)
83 \( 1 - 574. iT - 5.71e5T^{2} \)
89 \( 1 + 66.7iT - 7.04e5T^{2} \)
97 \( 1 - 1.43e3iT - 9.12e5T^{2} \)
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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

−13.49446368952579724867469745596, −12.44659639603886614438660368438, −10.49552246571708860951671917529, −9.496562976691881982089466116712, −8.257761145054209360831077569957, −7.34696308426532355518622749230, −6.58069158454424595187403275761, −4.97310403215082022596600585596, −4.21499655669178767476326044704, −0.37285511647387712587146729768, 2.06578689644116213817655823549, 2.97540791515606460855049990846, 4.59278120986566415526110661767, 6.03209058916779068237053169881, 8.263036900372385501786206717755, 9.222952740416132511091501245603, 10.11290978580980408788101653022, 11.21560084895045645001580920797, 12.05874448827510731504853080398, 12.55592694232754577263899609005

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