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Properties

Label 2-117-117.94-c1-0-7
Degree $2$
Conductor $117$
Sign $0.679 + 0.733i$
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  + (0.602 − 1.04i)2-s + (−1.54 + 0.788i)3-s + (0.274 + 0.476i)4-s + (1.89 − 3.27i)5-s + (−0.106 + 2.08i)6-s + 0.300·7-s + 3.07·8-s + (1.75 − 2.43i)9-s + (−2.27 − 3.94i)10-s + (−0.642 + 1.11i)11-s + (−0.799 − 0.517i)12-s + (−3.31 + 1.42i)13-s + (0.180 − 0.313i)14-s + (−0.333 + 6.54i)15-s + (1.29 − 2.25i)16-s + (−2.63 + 4.56i)17-s + ⋯
L(s)  = 1  + (0.425 − 0.737i)2-s + (−0.890 + 0.455i)3-s + (0.137 + 0.238i)4-s + (0.846 − 1.46i)5-s + (−0.0433 + 0.850i)6-s + 0.113·7-s + 1.08·8-s + (0.585 − 0.810i)9-s + (−0.720 − 1.24i)10-s + (−0.193 + 0.335i)11-s + (−0.230 − 0.149i)12-s + (−0.918 + 0.394i)13-s + (0.0483 − 0.0837i)14-s + (−0.0861 + 1.68i)15-s + (0.324 − 0.562i)16-s + (−0.639 + 1.10i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.679 + 0.733i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.679 + 0.733i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(117\)    =    \(3^{2} \cdot 13\)
Sign: $0.679 + 0.733i$
Analytic conductor: \(0.934249\)
Root analytic conductor: \(0.966565\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{117} (94, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 117,\ (\ :1/2),\ 0.679 + 0.733i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.09688 - 0.479463i\)
\(L(\frac12)\) \(\approx\) \(1.09688 - 0.479463i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.54 - 0.788i)T \)
13 \( 1 + (3.31 - 1.42i)T \)
good2 \( 1 + (-0.602 + 1.04i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 + (-1.89 + 3.27i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 - 0.300T + 7T^{2} \)
11 \( 1 + (0.642 - 1.11i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (2.63 - 4.56i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.829 - 1.43i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + 3.34T + 23T^{2} \)
29 \( 1 + (-4.81 + 8.34i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (3.29 - 5.71i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1.97 - 3.42i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 2.86T + 41T^{2} \)
43 \( 1 + 0.0208T + 43T^{2} \)
47 \( 1 + (-0.954 - 1.65i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 7.15T + 53T^{2} \)
59 \( 1 + (4.36 + 7.56i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 - 5.13T + 61T^{2} \)
67 \( 1 - 8.31T + 67T^{2} \)
71 \( 1 + (-4.64 + 8.03i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + 4.69T + 73T^{2} \)
79 \( 1 + (-6.65 - 11.5i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (6.45 + 11.1i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-3.19 - 5.54i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 3.76T + 97T^{2} \)
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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

−12.82407002170517515528318786705, −12.51279110101184115222182364572, −11.55577485214670713905627475655, −10.33664635562639490140654795396, −9.544204175497295497816605956212, −8.127683429415816687552642823064, −6.35288860106198118873228556418, −4.98458042714141340039622862644, −4.25204181537777910375218626319, −1.84337873332427114977916967759, 2.36930153080129256176389279579, 4.97282338417640281912927747292, 5.97158360607860060507307379422, 6.82831642156906060147661831081, 7.52535346946499569560748206724, 9.818079445174680622180795458007, 10.69363334835718735991837114827, 11.36970532903384492988146875599, 12.91411974129902714982462974219, 13.92018510801060379968052741099

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