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Properties

Label 2-117-117.103-c3-0-15
Degree $2$
Conductor $117$
Sign $0.703 - 0.711i$
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  + (1.73 − 1.00i)2-s + (−3.28 − 4.02i)3-s + (−1.99 + 3.44i)4-s + (16.4 + 9.47i)5-s + (−9.73 − 3.69i)6-s + (−21.8 + 12.6i)7-s + 24.0i·8-s + (−5.39 + 26.4i)9-s + 38.0·10-s + (35.8 − 20.7i)11-s + (20.4 − 3.32i)12-s + (−32.0 + 34.1i)13-s + (−25.2 + 43.7i)14-s + (−15.8 − 97.2i)15-s + (8.14 + 14.1i)16-s + 126.·17-s + ⋯
L(s)  = 1  + (0.613 − 0.354i)2-s + (−0.632 − 0.774i)3-s + (−0.248 + 0.431i)4-s + (1.46 + 0.847i)5-s + (−0.662 − 0.251i)6-s + (−1.17 + 0.680i)7-s + 1.06i·8-s + (−0.199 + 0.979i)9-s + 1.20·10-s + (0.983 − 0.567i)11-s + (0.491 − 0.0799i)12-s + (−0.684 + 0.729i)13-s + (−0.482 + 0.835i)14-s + (−0.272 − 1.67i)15-s + (0.127 + 0.220i)16-s + 1.80·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.703 - 0.711i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.703 - 0.711i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(117\)    =    \(3^{2} \cdot 13\)
Sign: $0.703 - 0.711i$
Analytic conductor: \(6.90322\)
Root analytic conductor: \(2.62739\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{117} (103, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 117,\ (\ :3/2),\ 0.703 - 0.711i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.62585 + 0.678756i\)
\(L(\frac12)\) \(\approx\) \(1.62585 + 0.678756i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (3.28 + 4.02i)T \)
13 \( 1 + (32.0 - 34.1i)T \)
good2 \( 1 + (-1.73 + 1.00i)T + (4 - 6.92i)T^{2} \)
5 \( 1 + (-16.4 - 9.47i)T + (62.5 + 108. i)T^{2} \)
7 \( 1 + (21.8 - 12.6i)T + (171.5 - 297. i)T^{2} \)
11 \( 1 + (-35.8 + 20.7i)T + (665.5 - 1.15e3i)T^{2} \)
17 \( 1 - 126.T + 4.91e3T^{2} \)
19 \( 1 + 10.9iT - 6.85e3T^{2} \)
23 \( 1 + (65.9 - 114. i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (52.0 + 90.2i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (126. + 73.2i)T + (1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 - 48.0iT - 5.06e4T^{2} \)
41 \( 1 + (-22.6 - 13.0i)T + (3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (32.2 + 55.8i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (-119. + 69.2i)T + (5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 - 221.T + 1.48e5T^{2} \)
59 \( 1 + (-115. - 66.8i)T + (1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-300. - 519. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-283. - 163. i)T + (1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 944. iT - 3.57e5T^{2} \)
73 \( 1 + 503. iT - 3.89e5T^{2} \)
79 \( 1 + (123. + 213. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (-491. + 283. i)T + (2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 + 480. iT - 7.04e5T^{2} \)
97 \( 1 + (-589. + 340. i)T + (4.56e5 - 7.90e5i)T^{2} \)
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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

−13.20392477162105267102027899413, −12.19799726618034605495794639660, −11.55686138433996461856564012976, −10.04952195239179566758785769017, −9.165084885121273843350073792511, −7.36448924073252088653339065666, −6.13542437231207724867390116288, −5.56815326572532188417679097181, −3.32309587548588124401375910574, −2.08214500302914246663302581669, 0.882278362461893055737972528345, 3.75882251005147385502851671460, 5.08353287661804276282941878511, 5.83495921827130037247071085627, 6.77293437161179177115149172640, 9.207139529638823170899404982170, 9.956127149116693268739720736837, 10.20845234859633326206548984358, 12.46822559922760333460579829816, 12.77149547855258721713533733903

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