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Properties

Label 2-117-13.3-c3-0-9
Degree $2$
Conductor $117$
Sign $0.477 - 0.878i$
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  + (1.5 + 2.59i)2-s + (−0.5 + 0.866i)4-s + 9·5-s + (−1 + 1.73i)7-s + 21·8-s + (13.5 + 23.3i)10-s + (15 + 25.9i)11-s + (32.5 − 33.7i)13-s − 6·14-s + (35.5 + 61.4i)16-s + (−55.5 + 96.1i)17-s + (23 − 39.8i)19-s + (−4.5 + 7.79i)20-s + (−45 + 77.9i)22-s + (−3 − 5.19i)23-s + ⋯
L(s)  = 1  + (0.530 + 0.918i)2-s + (−0.0625 + 0.108i)4-s + 0.804·5-s + (−0.0539 + 0.0935i)7-s + 0.928·8-s + (0.426 + 0.739i)10-s + (0.411 + 0.712i)11-s + (0.693 − 0.720i)13-s − 0.114·14-s + (0.554 + 0.960i)16-s + (−0.791 + 1.37i)17-s + (0.277 − 0.481i)19-s + (−0.0503 + 0.0871i)20-s + (−0.436 + 0.755i)22-s + (−0.0271 − 0.0471i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.477 - 0.878i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.477 - 0.878i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(117\)    =    \(3^{2} \cdot 13\)
Sign: $0.477 - 0.878i$
Analytic conductor: \(6.90322\)
Root analytic conductor: \(2.62739\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{117} (55, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 117,\ (\ :3/2),\ 0.477 - 0.878i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.22183 + 1.32105i\)
\(L(\frac12)\) \(\approx\) \(2.22183 + 1.32105i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-32.5 + 33.7i)T \)
good2 \( 1 + (-1.5 - 2.59i)T + (-4 + 6.92i)T^{2} \)
5 \( 1 - 9T + 125T^{2} \)
7 \( 1 + (1 - 1.73i)T + (-171.5 - 297. i)T^{2} \)
11 \( 1 + (-15 - 25.9i)T + (-665.5 + 1.15e3i)T^{2} \)
17 \( 1 + (55.5 - 96.1i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-23 + 39.8i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (3 + 5.19i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (52.5 + 90.9i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + 100T + 2.97e4T^{2} \)
37 \( 1 + (8.5 + 14.7i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + (115.5 + 200. i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (-257 + 445. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 - 162T + 1.03e5T^{2} \)
53 \( 1 + 639T + 1.48e5T^{2} \)
59 \( 1 + (-300 + 519. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (116.5 - 201. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (463 + 801. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + (465 - 805. i)T + (-1.78e5 - 3.09e5i)T^{2} \)
73 \( 1 + 253T + 3.89e5T^{2} \)
79 \( 1 + 1.32e3T + 4.93e5T^{2} \)
83 \( 1 + 810T + 5.71e5T^{2} \)
89 \( 1 + (-249 - 431. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (679 - 1.17e3i)T + (-4.56e5 - 7.90e5i)T^{2} \)
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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

−13.43210084596291737933072821190, −12.63580747924479550990377985281, −11.00456295744853967184249723267, −10.10679819391296597530508796590, −8.824106990181890564379696977260, −7.46009440396919559274862177859, −6.30277198921207241574782966663, −5.56085498300444515836785084161, −4.11761748298800714492538725836, −1.83443535941580837755260848413, 1.56714298362230859634875059950, 3.04725332939264108393957179846, 4.41192177725563983346359950072, 5.91482495112269628691916948868, 7.24866087342947947889409969194, 8.855199398443890316467726803264, 9.889836037190048215400670228634, 11.13650096035384196506808555589, 11.67303944268787283790913006182, 12.98181605368131781045299577396

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