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Properties

Label 2-117-39.2-c1-0-3
Degree $2$
Conductor $117$
Sign $-0.545 + 0.838i$
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  + (0.517 − 1.93i)2-s + (−1.73 − 0.999i)4-s + (−1.93 − 1.93i)5-s + (−0.5 + 0.133i)7-s + (−4.73 + 2.73i)10-s + (4.05 + 1.08i)11-s + (3.59 − 0.232i)13-s + 1.03i·14-s + (−1.99 − 3.46i)16-s + (−3.34 + 5.79i)17-s + (1.63 + 6.09i)19-s + (1.41 + 5.27i)20-s + (4.19 − 7.26i)22-s + (−1.22 − 2.12i)23-s + 2.46i·25-s + (1.41 − 7.07i)26-s + ⋯
L(s)  = 1  + (0.366 − 1.36i)2-s + (−0.866 − 0.499i)4-s + (−0.863 − 0.863i)5-s + (−0.188 + 0.0506i)7-s + (−1.49 + 0.863i)10-s + (1.22 + 0.327i)11-s + (0.997 − 0.0643i)13-s + 0.276i·14-s + (−0.499 − 0.866i)16-s + (−0.811 + 1.40i)17-s + (0.374 + 1.39i)19-s + (0.316 + 1.18i)20-s + (0.894 − 1.54i)22-s + (−0.255 − 0.442i)23-s + 0.492i·25-s + (0.277 − 1.38i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.545 + 0.838i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.545 + 0.838i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(117\)    =    \(3^{2} \cdot 13\)
Sign: $-0.545 + 0.838i$
Analytic conductor: \(0.934249\)
Root analytic conductor: \(0.966565\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{117} (80, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 117,\ (\ :1/2),\ -0.545 + 0.838i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.551865 - 1.01751i\)
\(L(\frac12)\) \(\approx\) \(0.551865 - 1.01751i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.59 + 0.232i)T \)
good2 \( 1 + (-0.517 + 1.93i)T + (-1.73 - i)T^{2} \)
5 \( 1 + (1.93 + 1.93i)T + 5iT^{2} \)
7 \( 1 + (0.5 - 0.133i)T + (6.06 - 3.5i)T^{2} \)
11 \( 1 + (-4.05 - 1.08i)T + (9.52 + 5.5i)T^{2} \)
17 \( 1 + (3.34 - 5.79i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.63 - 6.09i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (1.22 + 2.12i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.22 - 0.707i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-4.63 + 4.63i)T - 31iT^{2} \)
37 \( 1 + (-0.830 + 3.09i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + (0.378 - 1.41i)T + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (6.69 + 3.86i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (2.31 - 2.31i)T - 47iT^{2} \)
53 \( 1 - 5.93iT - 53T^{2} \)
59 \( 1 + (-0.0507 - 0.189i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (6.59 - 11.4i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (7.59 + 2.03i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + (-15.1 + 4.05i)T + (61.4 - 35.5i)T^{2} \)
73 \( 1 + (-2.90 - 2.90i)T + 73iT^{2} \)
79 \( 1 - 7.19T + 79T^{2} \)
83 \( 1 + (5.27 + 5.27i)T + 83iT^{2} \)
89 \( 1 + (-4.38 - 1.17i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (1.30 + 4.86i)T + (-84.0 + 48.5i)T^{2} \)
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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

−12.76367935723872927083991288945, −12.16246941996678712232031188835, −11.37838120707858518717075245438, −10.31144438551300015823765486649, −9.097027627913142309941748112587, −8.053913301041704665822532737203, −6.26823562621844535330467014374, −4.33220781145468761107111636161, −3.71693415295866371271252561843, −1.49998061370157750183068441136, 3.45270564034734209378757651124, 4.84011772685990671517079182167, 6.55401827770154978948195269383, 6.90004683780774711840549294734, 8.180019746373679788349659456466, 9.291309914553960617285800502080, 11.13928500468231888203781132758, 11.58970845904168415741649240002, 13.45188744278434239029895447285, 14.00633079736634740634376115752

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