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Properties

Label 2-117-117.103-c1-0-11
Degree $2$
Conductor $117$
Sign $-0.994 - 0.106i$
Analytic cond. $0.934249$
Root an. cond. $0.966565$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.5 − 0.866i)3-s + (−1 + 1.73i)4-s + (−3 − 1.73i)5-s + (−3 + 1.73i)7-s + (1.5 + 2.59i)9-s + (3 − 1.73i)11-s + (3 − 1.73i)12-s + (−2.5 − 2.59i)13-s + (3 + 5.19i)15-s + (−1.99 − 3.46i)16-s − 3·17-s + 3.46i·19-s + (6 − 3.46i)20-s + 6·21-s + (−1.5 + 2.59i)23-s + ⋯
L(s)  = 1  + (−0.866 − 0.499i)3-s + (−0.5 + 0.866i)4-s + (−1.34 − 0.774i)5-s + (−1.13 + 0.654i)7-s + (0.5 + 0.866i)9-s + (0.904 − 0.522i)11-s + (0.866 − 0.499i)12-s + (−0.693 − 0.720i)13-s + (0.774 + 1.34i)15-s + (−0.499 − 0.866i)16-s − 0.727·17-s + 0.794i·19-s + (1.34 − 0.774i)20-s + 1.30·21-s + (−0.312 + 0.541i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.994 - 0.106i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.994 - 0.106i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(117\)    =    \(3^{2} \cdot 13\)
Sign: $-0.994 - 0.106i$
Analytic conductor: \(0.934249\)
Root analytic conductor: \(0.966565\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{117} (103, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 117,\ (\ :1/2),\ -0.994 - 0.106i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.5 + 0.866i)T \)
13 \( 1 + (2.5 + 2.59i)T \)
good2 \( 1 + (1 - 1.73i)T^{2} \)
5 \( 1 + (3 + 1.73i)T + (2.5 + 4.33i)T^{2} \)
7 \( 1 + (3 - 1.73i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-3 + 1.73i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + 3T + 17T^{2} \)
19 \( 1 - 3.46iT - 19T^{2} \)
23 \( 1 + (1.5 - 2.59i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (3 + 5.19i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-3 - 1.73i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 - 6.92iT - 37T^{2} \)
41 \( 1 + (6 + 3.46i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.5 - 0.866i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (6 - 3.46i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + 9T + 53T^{2} \)
59 \( 1 + (-3 - 1.73i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.5 + 6.06i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 + 10.3iT - 73T^{2} \)
79 \( 1 + (0.5 + 0.866i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-6 + 3.46i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 - 3.46iT - 89T^{2} \)
97 \( 1 + (-6 + 3.46i)T + (48.5 - 84.0i)T^{2} \)
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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

−12.64976601342064648747269327247, −12.12900601149315251790140746565, −11.48412798336320981215192535822, −9.681905834105698325070932089493, −8.477662955428252514888537980504, −7.63110419811954353490859704665, −6.28951309323056334665448752408, −4.76331302955890611710624149097, −3.47613086007436455330254655724, 0, 3.80247328858458021016633500459, 4.64371638059730855232114849506, 6.54728851093017236777411448367, 6.99488438903327558810410830610, 9.144995044936558140044674438003, 10.00178610684291064617627635800, 10.94226950350821712296634159637, 11.75438835735683116415305978526, 12.90540293630586733890153355394

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