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Properties

Label 2-117-9.5-c2-0-23
Degree $2$
Conductor $117$
Sign $-0.775 + 0.631i$
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  + (2.73 − 1.58i)2-s + (−2.89 − 0.796i)3-s + (2.99 − 5.19i)4-s + (−6.90 − 3.98i)5-s + (−9.17 + 2.39i)6-s + (−1.88 − 3.27i)7-s − 6.30i·8-s + (7.73 + 4.60i)9-s − 25.2·10-s + (15.3 − 8.84i)11-s + (−12.8 + 12.6i)12-s + (1.80 − 3.12i)13-s + (−10.3 − 5.97i)14-s + (16.8 + 17.0i)15-s + (2.01 + 3.49i)16-s − 6.05i·17-s + ⋯
L(s)  = 1  + (1.36 − 0.790i)2-s + (−0.964 − 0.265i)3-s + (0.749 − 1.29i)4-s + (−1.38 − 0.797i)5-s + (−1.52 + 0.398i)6-s + (−0.269 − 0.467i)7-s − 0.788i·8-s + (0.858 + 0.512i)9-s − 2.52·10-s + (1.39 − 0.804i)11-s + (−1.06 + 1.05i)12-s + (0.138 − 0.240i)13-s + (−0.738 − 0.426i)14-s + (1.12 + 1.13i)15-s + (0.126 + 0.218i)16-s − 0.356i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.775 + 0.631i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.775 + 0.631i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(117\)    =    \(3^{2} \cdot 13\)
Sign: $-0.775 + 0.631i$
Analytic conductor: \(3.18801\)
Root analytic conductor: \(1.78550\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{117} (14, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 117,\ (\ :1),\ -0.775 + 0.631i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.550510 - 1.54624i\)
\(L(\frac12)\) \(\approx\) \(0.550510 - 1.54624i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (2.89 + 0.796i)T \)
13 \( 1 + (-1.80 + 3.12i)T \)
good2 \( 1 + (-2.73 + 1.58i)T + (2 - 3.46i)T^{2} \)
5 \( 1 + (6.90 + 3.98i)T + (12.5 + 21.6i)T^{2} \)
7 \( 1 + (1.88 + 3.27i)T + (-24.5 + 42.4i)T^{2} \)
11 \( 1 + (-15.3 + 8.84i)T + (60.5 - 104. i)T^{2} \)
17 \( 1 + 6.05iT - 289T^{2} \)
19 \( 1 - 13.2T + 361T^{2} \)
23 \( 1 + (24.8 + 14.3i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + (-18.0 + 10.4i)T + (420.5 - 728. i)T^{2} \)
31 \( 1 + (21.5 - 37.2i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 - 1.44T + 1.36e3T^{2} \)
41 \( 1 + (13.0 + 7.55i)T + (840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (-21.7 - 37.6i)T + (-924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (-34.1 + 19.7i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + 77.4iT - 2.80e3T^{2} \)
59 \( 1 + (-79.5 - 45.9i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-12.3 - 21.4i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (63.4 - 109. i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 79.9iT - 5.04e3T^{2} \)
73 \( 1 - 15.5T + 5.32e3T^{2} \)
79 \( 1 + (50.7 + 87.9i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (-11.0 + 6.38i)T + (3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 + 127. iT - 7.92e3T^{2} \)
97 \( 1 + (-13.3 - 23.1i)T + (-4.70e3 + 8.14e3i)T^{2} \)
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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

−12.63420729080294916369187948071, −11.76501267831768257687956514148, −11.58001575817014546822734064670, −10.32442016109587989941354367696, −8.483482605916413829736180486136, −6.97871357361311155095446160961, −5.67054207039197744346902095299, −4.42645269970873825266412866247, −3.66340202815230792171238206527, −0.944237323125544040478075302942, 3.66886604993294519558202294254, 4.33212105456582657684338852889, 5.85199427373986461253086149018, 6.77660416126717208033120814367, 7.56860590809157225856054059862, 9.566775760342048497191516765682, 11.10326807782303440323387887359, 12.10384978769465358965680144769, 12.26229178351239554618603004240, 13.92842535145016567514886880899

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