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Properties

Label 2-117-13.6-c2-0-8
Degree $2$
Conductor $117$
Sign $0.884 - 0.466i$
Analytic cond. $3.18801$
Root an. cond. $1.78550$
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  + (0.5 + 1.86i)2-s + (0.232 − 0.133i)4-s + (4.36 − 4.36i)5-s + (2.26 − 8.46i)7-s + (5.83 + 5.83i)8-s + (10.3 + 5.96i)10-s + (−6.19 + 1.66i)11-s + (−6.5 + 11.2i)13-s + 16.9·14-s + (−7.42 + 12.8i)16-s + (−9.99 + 5.76i)17-s + (3.36 + 0.901i)19-s + (0.428 − 1.59i)20-s + (−6.19 − 10.7i)22-s + (8.49 + 4.90i)23-s + ⋯
L(s)  = 1  + (0.250 + 0.933i)2-s + (0.0580 − 0.0334i)4-s + (0.873 − 0.873i)5-s + (0.323 − 1.20i)7-s + (0.728 + 0.728i)8-s + (1.03 + 0.596i)10-s + (−0.563 + 0.150i)11-s + (−0.5 + 0.866i)13-s + 1.20·14-s + (−0.464 + 0.804i)16-s + (−0.587 + 0.339i)17-s + (0.177 + 0.0474i)19-s + (0.0214 − 0.0799i)20-s + (−0.281 − 0.487i)22-s + (0.369 + 0.213i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.884 - 0.466i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.884 - 0.466i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(117\)    =    \(3^{2} \cdot 13\)
Sign: $0.884 - 0.466i$
Analytic conductor: \(3.18801\)
Root analytic conductor: \(1.78550\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{117} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 117,\ (\ :1),\ 0.884 - 0.466i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.85976 + 0.460030i\)
\(L(\frac12)\) \(\approx\) \(1.85976 + 0.460030i\)
\(L(2)\) 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 \( 1 + (6.5 - 11.2i)T \)
good2 \( 1 + (-0.5 - 1.86i)T + (-3.46 + 2i)T^{2} \)
5 \( 1 + (-4.36 + 4.36i)T - 25iT^{2} \)
7 \( 1 + (-2.26 + 8.46i)T + (-42.4 - 24.5i)T^{2} \)
11 \( 1 + (6.19 - 1.66i)T + (104. - 60.5i)T^{2} \)
17 \( 1 + (9.99 - 5.76i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (-3.36 - 0.901i)T + (312. + 180.5i)T^{2} \)
23 \( 1 + (-8.49 - 4.90i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + (5.69 - 9.86i)T + (-420.5 - 728. i)T^{2} \)
31 \( 1 + (-1.92 + 1.92i)T - 961iT^{2} \)
37 \( 1 + (42.1 - 11.2i)T + (1.18e3 - 684.5i)T^{2} \)
41 \( 1 + (5.08 + 18.9i)T + (-1.45e3 + 840.5i)T^{2} \)
43 \( 1 + (-45 + 25.9i)T + (924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (0.320 + 0.320i)T + 2.20e3iT^{2} \)
53 \( 1 + 78.7T + 2.80e3T^{2} \)
59 \( 1 + (10.9 - 40.9i)T + (-3.01e3 - 1.74e3i)T^{2} \)
61 \( 1 + (49.1 + 85.1i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-19.9 - 74.5i)T + (-3.88e3 + 2.24e3i)T^{2} \)
71 \( 1 + (-31.0 - 8.31i)T + (4.36e3 + 2.52e3i)T^{2} \)
73 \( 1 + (-48.2 - 48.2i)T + 5.32e3iT^{2} \)
79 \( 1 + 82.7T + 6.24e3T^{2} \)
83 \( 1 + (-69.5 + 69.5i)T - 6.88e3iT^{2} \)
89 \( 1 + (-31.8 + 8.52i)T + (6.85e3 - 3.96e3i)T^{2} \)
97 \( 1 + (-74.8 - 20.0i)T + (8.14e3 + 4.70e3i)T^{2} \)
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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

−13.72476091100425639821470994967, −12.70089485289403244922890464251, −11.16260297646614537620239813661, −10.20251408990483604436756016377, −8.953550418840949620956976552447, −7.64452218802888765454604993187, −6.73221072594809962542770938333, −5.39693521225276615329407324784, −4.48601167121298465673405389167, −1.73552494048310208940632732094, 2.21734899732098843030985143830, 2.98660309311086061713674839472, 5.09527189466375724664663282836, 6.36730020975169760639902085143, 7.73971469795538567032374172500, 9.288510324567637433568674341859, 10.36647045301870142295350018769, 11.08966088582312802240054914470, 12.15886424512459801082979511783, 13.00663185896081728591841814558

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