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Properties

Label 2-117-117.16-c1-0-10
Degree $2$
Conductor $117$
Sign $0.860 + 0.509i$
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  + 2.27·2-s + (−1.33 − 1.10i)3-s + 3.19·4-s + (−1.46 − 2.54i)5-s + (−3.04 − 2.51i)6-s + (2.42 + 4.20i)7-s + 2.72·8-s + (0.571 + 2.94i)9-s + (−3.35 − 5.80i)10-s − 0.394·11-s + (−4.27 − 3.52i)12-s + (−3.51 + 0.812i)13-s + (5.53 + 9.59i)14-s + (−0.841 + 5.02i)15-s − 0.173·16-s + (−1.10 + 1.91i)17-s + ⋯
L(s)  = 1  + 1.61·2-s + (−0.771 − 0.636i)3-s + 1.59·4-s + (−0.657 − 1.13i)5-s + (−1.24 − 1.02i)6-s + (0.918 + 1.59i)7-s + 0.964·8-s + (0.190 + 0.981i)9-s + (−1.05 − 1.83i)10-s − 0.119·11-s + (−1.23 − 1.01i)12-s + (−0.974 + 0.225i)13-s + (1.48 + 2.56i)14-s + (−0.217 + 1.29i)15-s − 0.0433·16-s + (−0.267 + 0.464i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.860 + 0.509i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.860 + 0.509i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(117\)    =    \(3^{2} \cdot 13\)
Sign: $0.860 + 0.509i$
Analytic conductor: \(0.934249\)
Root analytic conductor: \(0.966565\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{117} (16, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 117,\ (\ :1/2),\ 0.860 + 0.509i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.70934 - 0.468195i\)
\(L(\frac12)\) \(\approx\) \(1.70934 - 0.468195i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.33 + 1.10i)T \)
13 \( 1 + (3.51 - 0.812i)T \)
good2 \( 1 - 2.27T + 2T^{2} \)
5 \( 1 + (1.46 + 2.54i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (-2.42 - 4.20i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + 0.394T + 11T^{2} \)
17 \( 1 + (1.10 - 1.91i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.45 + 2.52i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.14 + 3.71i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 1.53T + 29T^{2} \)
31 \( 1 + (1.98 + 3.44i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1.39 - 2.42i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-1.04 + 1.80i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-1.07 - 1.85i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (0.294 - 0.510i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 7.61T + 53T^{2} \)
59 \( 1 - 2.63T + 59T^{2} \)
61 \( 1 + (-0.561 - 0.972i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-4.82 + 8.35i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-0.232 + 0.402i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 - 2.12T + 73T^{2} \)
79 \( 1 + (-7.09 + 12.2i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (1.30 - 2.26i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (3.71 + 6.43i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-7.48 - 12.9i)T + (-48.5 + 84.0i)T^{2} \)
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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

−13.04566965921906967572847595986, −12.43013524565529146966867979561, −11.88912191197753755763509974539, −11.18361890773344638612505517490, −8.927038083251275879359411051401, −7.76564860786090147211988750411, −6.24500446439411197557638209284, −5.07504495953605410577137965824, −4.71362616055364535978158605705, −2.29231573255789849666141579840, 3.33954195128548369005454272387, 4.30672196351173367863295409675, 5.26272104941540121799838701079, 6.83702976445499522442059898242, 7.47746240686503719761655632197, 10.03404619992297855168575322376, 11.05280926438510328427881532110, 11.40986528598049467294542930517, 12.55841648860208322040168152743, 13.91533904949584779196672563027

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