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Properties

Label 2-117-9.4-c1-0-11
Degree $2$
Conductor $117$
Sign $-0.173 - 0.984i$
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 − 1.73i)2-s + (−1.5 − 0.866i)3-s + (−0.999 + 1.73i)4-s + (−2 + 3.46i)5-s + 3.46i·6-s + (−1 − 1.73i)7-s + (1.5 + 2.59i)9-s + 7.99·10-s + (−1 − 1.73i)11-s + (2.99 − 1.73i)12-s + (−0.5 + 0.866i)13-s + (−1.99 + 3.46i)14-s + (6 − 3.46i)15-s + (1.99 + 3.46i)16-s − 5·17-s + (3 − 5.19i)18-s + ⋯
L(s)  = 1  + (−0.707 − 1.22i)2-s + (−0.866 − 0.499i)3-s + (−0.499 + 0.866i)4-s + (−0.894 + 1.54i)5-s + 1.41i·6-s + (−0.377 − 0.654i)7-s + (0.5 + 0.866i)9-s + 2.52·10-s + (−0.301 − 0.522i)11-s + (0.866 − 0.500i)12-s + (−0.138 + 0.240i)13-s + (−0.534 + 0.925i)14-s + (1.54 − 0.894i)15-s + (0.499 + 0.866i)16-s − 1.21·17-s + (0.707 − 1.22i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.173 - 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.173 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(117\)    =    \(3^{2} \cdot 13\)
Sign: $-0.173 - 0.984i$
Analytic conductor: \(0.934249\)
Root analytic conductor: \(0.966565\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{117} (40, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 117,\ (\ :1/2),\ -0.173 - 0.984i)\)

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 \( 1 + (0.5 - 0.866i)T \)
good2 \( 1 + (1 + 1.73i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + (2 - 3.46i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (1 + 1.73i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (1 + 1.73i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + 5T + 17T^{2} \)
19 \( 1 + 6T + 19T^{2} \)
23 \( 1 + (-1.5 + 2.59i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1 - 1.73i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (2 - 3.46i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 2T + 37T^{2} \)
41 \( 1 + (-3 + 5.19i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-1.5 - 2.59i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (3 + 5.19i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 3T + 53T^{2} \)
59 \( 1 + (6 - 10.3i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.5 - 4.33i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (2 - 3.46i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 12T + 71T^{2} \)
73 \( 1 + 4T + 73T^{2} \)
79 \( 1 + (2.5 + 4.33i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (1 + 1.73i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 10T + 89T^{2} \)
97 \( 1 + (-4 - 6.92i)T + (-48.5 + 84.0i)T^{2} \)
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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

−12.36886985559743262914532758057, −11.38179457796890927224920857107, −10.71820635543410582069665704574, −10.39039528355302913521713267168, −8.563269284622916452625217341821, −7.16409432509687099870075453860, −6.39603500228795107885277515346, −3.99955899920403288461230269357, −2.50964273036828341017701981849, 0, 4.36470287146865125139193462229, 5.38590008515613444261790887688, 6.54890166747572833456200045986, 7.905208079325265537173997537277, 8.891939893687426618524028996333, 9.567657908037077911008199632307, 11.19543785121556848574998822614, 12.34412890230565534100741464708, 12.88404769380007274792023612784

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