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Properties

Label 2-117-13.9-c1-0-1
Degree $2$
Conductor $117$
Sign $0.859 - 0.511i$
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  + (1 + 1.73i)4-s + (0.5 + 0.866i)7-s + (2.5 − 2.59i)13-s + (−1.99 + 3.46i)16-s + (−4 − 6.92i)19-s − 5·25-s + (−0.999 + 1.73i)28-s − 7·31-s + (5 − 8.66i)37-s + (6.5 + 11.2i)43-s + (3 − 5.19i)49-s + (7 + 1.73i)52-s + (6.5 + 11.2i)61-s − 7.99·64-s + (−5.5 + 9.52i)67-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)4-s + (0.188 + 0.327i)7-s + (0.693 − 0.720i)13-s + (−0.499 + 0.866i)16-s + (−0.917 − 1.58i)19-s − 25-s + (−0.188 + 0.327i)28-s − 1.25·31-s + (0.821 − 1.42i)37-s + (0.991 + 1.71i)43-s + (0.428 − 0.742i)49-s + (0.970 + 0.240i)52-s + (0.832 + 1.44i)61-s − 0.999·64-s + (−0.671 + 1.16i)67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.859 - 0.511i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.859 - 0.511i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(117\)    =    \(3^{2} \cdot 13\)
Sign: $0.859 - 0.511i$
Analytic conductor: \(0.934249\)
Root analytic conductor: \(0.966565\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{117} (100, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 117,\ (\ :1/2),\ 0.859 - 0.511i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.11090 + 0.305314i\)
\(L(\frac12)\) \(\approx\) \(1.11090 + 0.305314i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.5 + 2.59i)T \)
good2 \( 1 + (-1 - 1.73i)T^{2} \)
5 \( 1 + 5T^{2} \)
7 \( 1 + (-0.5 - 0.866i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (4 + 6.92i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 7T + 31T^{2} \)
37 \( 1 + (-5 + 8.66i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-6.5 - 11.2i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 + 53T^{2} \)
59 \( 1 + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-6.5 - 11.2i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (5.5 - 9.52i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-35.5 + 61.4i)T^{2} \)
73 \( 1 - 17T + 73T^{2} \)
79 \( 1 + 13T + 79T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (2.5 + 4.33i)T + (-48.5 + 84.0i)T^{2} \)
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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

−13.29495144815058327557878512138, −12.71655998394923824990111930580, −11.46977469887847279077121922683, −10.82091562245990221907594748138, −9.175929646603444156515207466466, −8.197187863482219290864543145817, −7.12479528520181278145061038849, −5.81473512122248334257704625242, −4.06267316216209923172480215955, −2.52895160986709551773881672407, 1.80387006944753394195323284491, 4.02105013198232063605481217469, 5.65501082476237562525325714497, 6.63122969102480163207734998762, 7.969176601547087292910961275462, 9.359902462929009005981090779922, 10.42730372558088741522092921022, 11.21867273264163268591275018335, 12.30038021614666650701412544902, 13.68878978936650234687996714425

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