↑

Properties

Label 2-117-117.94-c1-0-6
Degree $2$
Conductor $117$
Sign $0.999 + 0.0259i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.108 + 0.187i)2-s + (0.455 − 1.67i)3-s + (0.976 + 1.69i)4-s + (−0.702 + 1.21i)5-s + (0.263 + 0.266i)6-s + 3.31·7-s − 0.855·8-s + (−2.58 − 1.52i)9-s + (−0.151 − 0.263i)10-s + (2.10 − 3.64i)11-s + (3.27 − 0.860i)12-s + (−3.23 − 1.59i)13-s + (−0.358 + 0.621i)14-s + (1.71 + 1.72i)15-s + (−1.86 + 3.22i)16-s + (−2.86 + 4.96i)17-s + ⋯
L(s)  = 1  + (−0.0764 + 0.132i)2-s + (0.263 − 0.964i)3-s + (0.488 + 0.845i)4-s + (−0.314 + 0.543i)5-s + (0.107 + 0.108i)6-s + 1.25·7-s − 0.302·8-s + (−0.861 − 0.507i)9-s + (−0.0480 − 0.0832i)10-s + (0.634 − 1.09i)11-s + (0.944 − 0.248i)12-s + (−0.896 − 0.443i)13-s + (−0.0958 + 0.165i)14-s + (0.442 + 0.446i)15-s + (−0.465 + 0.805i)16-s + (−0.694 + 1.20i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0259i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0259i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(117\)    =    \(3^{2} \cdot 13\)
Sign: $0.999 + 0.0259i$
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: \((2,\ 117,\ (\ :1/2),\ 0.999 + 0.0259i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.18534 - 0.0153589i\)
\(L(\frac12)\) \(\approx\) \(1.18534 - 0.0153589i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.455 + 1.67i)T \)
13 \( 1 + (3.23 + 1.59i)T \)
good2 \( 1 + (0.108 - 0.187i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 + (0.702 - 1.21i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 - 3.31T + 7T^{2} \)
11 \( 1 + (-2.10 + 3.64i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (2.86 - 4.96i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.190 + 0.330i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + 6.81T + 23T^{2} \)
29 \( 1 + (-2.28 + 3.95i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-0.165 + 0.287i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (4.85 + 8.40i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 1.11T + 41T^{2} \)
43 \( 1 + 2.10T + 43T^{2} \)
47 \( 1 + (-0.177 - 0.308i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 8.84T + 53T^{2} \)
59 \( 1 + (-6.70 - 11.6i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 - 0.890T + 61T^{2} \)
67 \( 1 - 0.781T + 67T^{2} \)
71 \( 1 + (-5.32 + 9.21i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 - 0.857T + 73T^{2} \)
79 \( 1 + (2.58 + 4.48i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-4.78 - 8.28i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-2.55 - 4.42i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 7.32T + 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−13.55454993071018236231495674450, −12.33062890709741617369378703209, −11.62267057880350204031737237286, −10.81815204422619184047713948944, −8.684657596936447243452145732447, −8.029687239873073204179469268564, −7.15548432967204904782075070372, −5.97828861253696239179607852650, −3.76623041712074670840135770010, −2.21683201713112421950650388365, 2.12162693632257182523395385949, 4.51007291039901987837074552123, 5.07924102946823598683797946733, 6.95084426844419343736646905142, 8.377920895856911946157795798523, 9.493902139165470135562943745611, 10.27182984434267060045756618075, 11.53600740138769608874306166176, 11.99547103961468761373354481673, 14.09217359073030558014337099428

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