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Properties

Label 2-117-13.12-c3-0-9
Degree $2$
Conductor $117$
Sign $1$
Analytic cond. $6.90322$
Root an. cond. $2.62739$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4.42i·2-s − 11.6·4-s − 20.3i·5-s − 15.9i·8-s + 90.2·10-s − 70.0i·11-s + 46.8·13-s − 22.1·16-s + 236. i·20-s + 310.·22-s − 290.·25-s + 207. i·26-s − 225. i·32-s − 325.·40-s − 486. i·41-s + ⋯
L(s)  = 1  + 1.56i·2-s − 1.45·4-s − 1.82i·5-s − 0.705i·8-s + 2.85·10-s − 1.91i·11-s + 1.00·13-s − 0.346·16-s + 2.64i·20-s + 3.00·22-s − 2.32·25-s + 1.56i·26-s − 1.24i·32-s − 1.28·40-s − 1.85i·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(117\)    =    \(3^{2} \cdot 13\)
Sign: $1$
Analytic conductor: \(6.90322\)
Root analytic conductor: \(2.62739\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{117} (64, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 117,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(1.33582\)
\(L(\frac12)\) \(\approx\) \(1.33582\)
\(L(\frac{5}{2})\) 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 \( 1 - 46.8T \)
good2 \( 1 - 4.42iT - 8T^{2} \)
5 \( 1 + 20.3iT - 125T^{2} \)
7 \( 1 - 343T^{2} \)
11 \( 1 + 70.0iT - 1.33e3T^{2} \)
17 \( 1 + 4.91e3T^{2} \)
19 \( 1 - 6.85e3T^{2} \)
23 \( 1 + 1.21e4T^{2} \)
29 \( 1 + 2.43e4T^{2} \)
31 \( 1 - 2.97e4T^{2} \)
37 \( 1 - 5.06e4T^{2} \)
41 \( 1 + 486. iT - 6.89e4T^{2} \)
43 \( 1 - 452T + 7.95e4T^{2} \)
47 \( 1 - 71.1iT - 1.03e5T^{2} \)
53 \( 1 + 1.48e5T^{2} \)
59 \( 1 - 696. iT - 2.05e5T^{2} \)
61 \( 1 + 944.T + 2.26e5T^{2} \)
67 \( 1 - 3.00e5T^{2} \)
71 \( 1 + 123. iT - 3.57e5T^{2} \)
73 \( 1 - 3.89e5T^{2} \)
79 \( 1 - 418.T + 4.93e5T^{2} \)
83 \( 1 - 1.50e3iT - 5.71e5T^{2} \)
89 \( 1 + 155. iT - 7.04e5T^{2} \)
97 \( 1 - 9.12e5T^{2} \)
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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

−13.59137040171676956124902723077, −12.33211395464503882824160919616, −11.00512185005866317058264490065, −8.973774865918991405321458958340, −8.735400857083690430728815360239, −7.74799160357728494602927486165, −6.02761560678818687588752145036, −5.46765207885993771344812385107, −4.08861906623796825448220164795, −0.74947780079364504799971857884, 1.93414730410789057295757879670, 3.07536055520435076079127533659, 4.28916520528875930588625783628, 6.43930566248715737132062517212, 7.52620405642343953344572689283, 9.408438562124640929267531943094, 10.24948645411412899446831703633, 10.91563164119839698012315134805, 11.78041628046654923804559932267, 12.80056317237265360561319787638

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