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Properties

Label 2-117-117.49-c1-0-10
Degree $2$
Conductor $117$
Sign $0.0469 + 0.998i$
Analytic cond. $0.934249$
Root an. cond. $0.966565$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.495 + 0.285i)2-s + (−1.06 − 1.36i)3-s + (−0.836 − 1.44i)4-s + (−0.796 − 0.459i)5-s + (−0.133 − 0.981i)6-s − 1.93i·7-s − 2.10i·8-s + (−0.751 + 2.90i)9-s + (−0.262 − 0.455i)10-s + (3.64 + 2.10i)11-s + (−1.09 + 2.68i)12-s + (1.35 − 3.34i)13-s + (0.552 − 0.957i)14-s + (0.214 + 1.57i)15-s + (−1.07 + 1.85i)16-s + (−1.20 + 2.08i)17-s + ⋯
L(s)  = 1  + (0.350 + 0.202i)2-s + (−0.612 − 0.790i)3-s + (−0.418 − 0.724i)4-s + (−0.356 − 0.205i)5-s + (−0.0544 − 0.400i)6-s − 0.730i·7-s − 0.742i·8-s + (−0.250 + 0.968i)9-s + (−0.0831 − 0.143i)10-s + (1.09 + 0.633i)11-s + (−0.316 + 0.774i)12-s + (0.375 − 0.926i)13-s + (0.147 − 0.255i)14-s + (0.0553 + 0.407i)15-s + (−0.268 + 0.464i)16-s + (−0.291 + 0.505i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0469 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0469 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(117\)    =    \(3^{2} \cdot 13\)
Sign: $0.0469 + 0.998i$
Analytic conductor: \(0.934249\)
Root analytic conductor: \(0.966565\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{117} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 117,\ (\ :1/2),\ 0.0469 + 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.656569 - 0.626463i\)
\(L(\frac12)\) \(\approx\) \(0.656569 - 0.626463i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.06 + 1.36i)T \)
13 \( 1 + (-1.35 + 3.34i)T \)
good2 \( 1 + (-0.495 - 0.285i)T + (1 + 1.73i)T^{2} \)
5 \( 1 + (0.796 + 0.459i)T + (2.5 + 4.33i)T^{2} \)
7 \( 1 + 1.93iT - 7T^{2} \)
11 \( 1 + (-3.64 - 2.10i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (1.20 - 2.08i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.60 - 0.928i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 - 8.22T + 23T^{2} \)
29 \( 1 + (2.36 - 4.09i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (4.29 + 2.47i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.959 - 0.554i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 - 0.566iT - 41T^{2} \)
43 \( 1 - 9.58T + 43T^{2} \)
47 \( 1 + (1.35 - 0.780i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + 4.09T + 53T^{2} \)
59 \( 1 + (-5.29 + 3.05i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 - 1.33T + 61T^{2} \)
67 \( 1 - 16.3iT - 67T^{2} \)
71 \( 1 + (10.6 + 6.12i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + 8.77iT - 73T^{2} \)
79 \( 1 + (-4.09 - 7.09i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (2.31 - 1.33i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (11.4 - 6.61i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 13.8iT - 97T^{2} \)
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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

−13.17830731193204968147651679879, −12.55063765140411391514066176698, −11.22713938458541825628975454490, −10.35393877213564706642984944461, −8.983369314072910311456652972084, −7.49795253107976412906088376305, −6.53050885999447860292698284118, −5.35603503100533143843514964735, −4.08111424099919375876892903888, −1.12192424498000645566301779800, 3.27629099499374492049080296653, 4.33924042196526366954104719069, 5.62519647458676794965417894053, 7.06474143637873597026215420957, 8.901218308659970390179310945596, 9.236011373954407537924878796611, 11.30286080502540533400108384054, 11.43855915762074736893149844158, 12.53648760097026639997745498018, 13.79709994179759040359933286242

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