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Properties

Label 2-117-9.2-c2-0-12
Degree $2$
Conductor $117$
Sign $0.752 + 0.658i$
Analytic cond. $3.18801$
Root an. cond. $1.78550$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.10 + 0.635i)2-s + (−2.90 + 0.746i)3-s + (−1.19 − 2.06i)4-s + (4.29 − 2.47i)5-s + (−3.67 − 1.02i)6-s + (2.07 − 3.59i)7-s − 8.11i·8-s + (7.88 − 4.34i)9-s + 6.30·10-s + (6.91 + 3.99i)11-s + (5.00 + 5.10i)12-s + (−1.80 − 3.12i)13-s + (4.57 − 2.64i)14-s + (−10.6 + 10.4i)15-s + (0.388 − 0.673i)16-s − 19.5i·17-s + ⋯
L(s)  = 1  + (0.550 + 0.317i)2-s + (−0.968 + 0.248i)3-s + (−0.298 − 0.516i)4-s + (0.858 − 0.495i)5-s + (−0.612 − 0.170i)6-s + (0.296 − 0.514i)7-s − 1.01i·8-s + (0.876 − 0.482i)9-s + 0.630·10-s + (0.628 + 0.362i)11-s + (0.417 + 0.425i)12-s + (−0.138 − 0.240i)13-s + (0.326 − 0.188i)14-s + (−0.708 + 0.693i)15-s + (0.0242 − 0.0420i)16-s − 1.14i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.752 + 0.658i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.752 + 0.658i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(117\)    =    \(3^{2} \cdot 13\)
Sign: $0.752 + 0.658i$
Analytic conductor: \(3.18801\)
Root analytic conductor: \(1.78550\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{117} (92, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 117,\ (\ :1),\ 0.752 + 0.658i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.34735 - 0.505962i\)
\(L(\frac12)\) \(\approx\) \(1.34735 - 0.505962i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (2.90 - 0.746i)T \)
13 \( 1 + (1.80 + 3.12i)T \)
good2 \( 1 + (-1.10 - 0.635i)T + (2 + 3.46i)T^{2} \)
5 \( 1 + (-4.29 + 2.47i)T + (12.5 - 21.6i)T^{2} \)
7 \( 1 + (-2.07 + 3.59i)T + (-24.5 - 42.4i)T^{2} \)
11 \( 1 + (-6.91 - 3.99i)T + (60.5 + 104. i)T^{2} \)
17 \( 1 + 19.5iT - 289T^{2} \)
19 \( 1 - 5.83T + 361T^{2} \)
23 \( 1 + (8.95 - 5.17i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + (30.4 + 17.5i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + (-19.2 - 33.3i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 - 58.1T + 1.36e3T^{2} \)
41 \( 1 + (11.9 - 6.92i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (8.84 - 15.3i)T + (-924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (-67.0 - 38.7i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + 58.8iT - 2.80e3T^{2} \)
59 \( 1 + (35.0 - 20.2i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (27.1 - 46.9i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-13.5 - 23.4i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 - 1.77iT - 5.04e3T^{2} \)
73 \( 1 - 128.T + 5.32e3T^{2} \)
79 \( 1 + (44.7 - 77.4i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (-15.7 - 9.08i)T + (3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 + 84.2iT - 7.92e3T^{2} \)
97 \( 1 + (37.4 - 64.7i)T + (-4.70e3 - 8.14e3i)T^{2} \)
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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

−13.32143904915307726174180028773, −12.28034837440069237123453022988, −11.10717290301032196370551660674, −9.864958407197055635418847604057, −9.401757656530165004181693778958, −7.25235059359507096690457312697, −6.08073553639815848529847940877, −5.18905957539448758885280226002, −4.25038674099498925759634655249, −1.11565100373678170315268059661, 2.13098114588776212177529118565, 4.05898372984993684490149530029, 5.49447388383585520910665762763, 6.35162291282058244088297414316, 7.88213024152732822798377741742, 9.279714830074225644896335472127, 10.59516373451177988568548604947, 11.55981987170713068059208098015, 12.32531150816771816516267553126, 13.29409316175257485271544741916

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