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Properties

Label 2-117-117.25-c1-0-3
Degree $2$
Conductor $117$
Sign $0.385 - 0.922i$
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  + (2.14 + 1.23i)2-s + (−1.72 + 0.0890i)3-s + (2.06 + 3.58i)4-s + (0.771 − 0.445i)5-s + (−3.82 − 1.95i)6-s + (−0.850 − 0.491i)7-s + 5.29i·8-s + (2.98 − 0.308i)9-s + 2.20·10-s + (−2.49 − 1.44i)11-s + (−3.89 − 6.01i)12-s + (−3.41 − 1.14i)13-s + (−1.21 − 2.10i)14-s + (−1.29 + 0.839i)15-s + (−2.42 + 4.19i)16-s + 5.34·17-s + ⋯
L(s)  = 1  + (1.51 + 0.875i)2-s + (−0.998 + 0.0514i)3-s + (1.03 + 1.79i)4-s + (0.344 − 0.199i)5-s + (−1.56 − 0.796i)6-s + (−0.321 − 0.185i)7-s + 1.87i·8-s + (0.994 − 0.102i)9-s + 0.697·10-s + (−0.753 − 0.435i)11-s + (−1.12 − 1.73i)12-s + (−0.947 − 0.318i)13-s + (−0.325 − 0.563i)14-s + (−0.334 + 0.216i)15-s + (−0.605 + 1.04i)16-s + 1.29·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.385 - 0.922i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.385 - 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(117\)    =    \(3^{2} \cdot 13\)
Sign: $0.385 - 0.922i$
Analytic conductor: \(0.934249\)
Root analytic conductor: \(0.966565\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{117} (25, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 117,\ (\ :1/2),\ 0.385 - 0.922i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.40922 + 0.938450i\)
\(L(\frac12)\) \(\approx\) \(1.40922 + 0.938450i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.72 - 0.0890i)T \)
13 \( 1 + (3.41 + 1.14i)T \)
good2 \( 1 + (-2.14 - 1.23i)T + (1 + 1.73i)T^{2} \)
5 \( 1 + (-0.771 + 0.445i)T + (2.5 - 4.33i)T^{2} \)
7 \( 1 + (0.850 + 0.491i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (2.49 + 1.44i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 - 5.34T + 17T^{2} \)
19 \( 1 + 7.19iT - 19T^{2} \)
23 \( 1 + (-2.31 - 4.01i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.971 - 1.68i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (8.73 - 5.04i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 - 4.82iT - 37T^{2} \)
41 \( 1 + (2.39 - 1.38i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (2.45 - 4.25i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-3.82 - 2.20i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 - 6.30T + 53T^{2} \)
59 \( 1 + (-2.74 + 1.58i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.76 + 4.78i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.15 + 1.81i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 6.69iT - 71T^{2} \)
73 \( 1 + 9.33iT - 73T^{2} \)
79 \( 1 + (2.37 - 4.11i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-4.78 - 2.76i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 - 17.5iT - 89T^{2} \)
97 \( 1 + (0.213 + 0.123i)T + (48.5 + 84.0i)T^{2} \)
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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

−13.48139672516641389202311642644, −12.97609455236706868750917834232, −12.06522909169268896466291205425, −10.98633128462004338642768042396, −9.638258185231627356551331128076, −7.60740301532725961865432871417, −6.80154877953018790742543384412, −5.41274136841263335712468635347, −5.09529454885261917655112867936, −3.32331454542003528481934627020, 2.18429954315169107360635852950, 3.95799080245064928222463911627, 5.30308442319878733756674050975, 5.95545638890156674057079294959, 7.37850315703351970785999267324, 9.975290593671165224290819515469, 10.37076656093484690611254100403, 11.66304181645479617214426715246, 12.42305721335060319421509492702, 12.91469816392560748654831980626

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