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Properties

Label 2-117-13.3-c1-0-1
Degree $2$
Conductor $117$
Sign $0.872 - 0.488i$
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  + (0.5 + 0.866i)2-s + (0.500 − 0.866i)4-s + 5-s + (−1 + 1.73i)7-s + 3·8-s + (0.5 + 0.866i)10-s + (−1 − 1.73i)11-s + (−3.5 + 0.866i)13-s − 1.99·14-s + (0.500 + 0.866i)16-s + (−3.5 + 6.06i)17-s + (3 − 5.19i)19-s + (0.500 − 0.866i)20-s + (0.999 − 1.73i)22-s + (−3 − 5.19i)23-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (0.250 − 0.433i)4-s + 0.447·5-s + (−0.377 + 0.654i)7-s + 1.06·8-s + (0.158 + 0.273i)10-s + (−0.301 − 0.522i)11-s + (−0.970 + 0.240i)13-s − 0.534·14-s + (0.125 + 0.216i)16-s + (−0.848 + 1.47i)17-s + (0.688 − 1.19i)19-s + (0.111 − 0.193i)20-s + (0.213 − 0.369i)22-s + (−0.625 − 1.08i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.872 - 0.488i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.872 - 0.488i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(117\)    =    \(3^{2} \cdot 13\)
Sign: $0.872 - 0.488i$
Analytic conductor: \(0.934249\)
Root analytic conductor: \(0.966565\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{117} (55, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 117,\ (\ :1/2),\ 0.872 - 0.488i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.29709 + 0.338656i\)
\(L(\frac12)\) \(\approx\) \(1.29709 + 0.338656i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
13 \( 1 + (3.5 - 0.866i)T \)
good2 \( 1 + (-0.5 - 0.866i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 - T + 5T^{2} \)
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 \( 1 + (3.5 - 6.06i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3 + 5.19i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (3 + 5.19i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.5 + 0.866i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 4T + 31T^{2} \)
37 \( 1 + (0.5 + 0.866i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-4.5 - 7.79i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (3 - 5.19i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 6T + 47T^{2} \)
53 \( 1 - 9T + 53T^{2} \)
59 \( 1 + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (0.5 - 0.866i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1 - 1.73i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-3 + 5.19i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 - 11T + 73T^{2} \)
79 \( 1 + 4T + 79T^{2} \)
83 \( 1 - 14T + 83T^{2} \)
89 \( 1 + (7 + 12.1i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-1 + 1.73i)T + (-48.5 - 84.0i)T^{2} \)
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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

−13.71583373732554622146040930695, −12.87183096216739190185826885775, −11.55009927918403594947634959404, −10.42252617046725000405145553302, −9.435674583513402062223853553021, −8.080135195646394940869329121953, −6.63571917544642377846356517840, −5.88321458374189873299594450808, −4.63689197056435120495687654562, −2.36681532156722348274801924146, 2.28342814195701605346487158022, 3.77137715802230625281806667228, 5.20143778442319878796371214143, 7.00209494575348991887569003457, 7.77585513134230575802964807258, 9.623652786259858771580091256995, 10.28989878879188507205957631370, 11.61694792247288167269886733238, 12.32825400348559817062130598804, 13.48498487929867655058308897439

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