Properties

Label 2-117-13.10-c3-0-7
Degree $2$
Conductor $117$
Sign $0.887 - 0.460i$
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  + (−3.57 + 2.06i)2-s + (4.50 − 7.79i)4-s + 3.05i·5-s + (−5.78 − 3.34i)7-s + 4.12i·8-s + (−6.28 − 10.8i)10-s + (27.9 − 16.1i)11-s + (−22.1 − 41.3i)13-s + 27.5·14-s + (27.4 + 47.6i)16-s + (14.4 − 24.9i)17-s + (87.6 + 50.5i)19-s + (23.7 + 13.7i)20-s + (−66.4 + 115. i)22-s + (59.4 + 103. i)23-s + ⋯
L(s)  = 1  + (−1.26 + 0.728i)2-s + (0.562 − 0.974i)4-s + 0.272i·5-s + (−0.312 − 0.180i)7-s + 0.182i·8-s + (−0.198 − 0.344i)10-s + (0.765 − 0.441i)11-s + (−0.472 − 0.881i)13-s + 0.526·14-s + (0.429 + 0.744i)16-s + (0.205 − 0.356i)17-s + (1.05 + 0.610i)19-s + (0.265 + 0.153i)20-s + (−0.644 + 1.11i)22-s + (0.539 + 0.934i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.887 - 0.460i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.887 - 0.460i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(117\)    =    \(3^{2} \cdot 13\)
Sign: $0.887 - 0.460i$
Analytic conductor: \(6.90322\)
Root analytic conductor: \(2.62739\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{117} (10, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 117,\ (\ :3/2),\ 0.887 - 0.460i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.782014 + 0.190975i\)
\(L(\frac12)\) \(\approx\) \(0.782014 + 0.190975i\)
\(L(\frac{5}{2})\) 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 \( 1 + (22.1 + 41.3i)T \)
good2 \( 1 + (3.57 - 2.06i)T + (4 - 6.92i)T^{2} \)
5 \( 1 - 3.05iT - 125T^{2} \)
7 \( 1 + (5.78 + 3.34i)T + (171.5 + 297. i)T^{2} \)
11 \( 1 + (-27.9 + 16.1i)T + (665.5 - 1.15e3i)T^{2} \)
17 \( 1 + (-14.4 + 24.9i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-87.6 - 50.5i)T + (3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-59.4 - 103. i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (-80.0 - 138. i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + 38.0iT - 2.97e4T^{2} \)
37 \( 1 + (-283. + 163. i)T + (2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + (48.5 - 28.0i)T + (3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (-63.9 + 110. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + 517. iT - 1.03e5T^{2} \)
53 \( 1 - 695.T + 1.48e5T^{2} \)
59 \( 1 + (568. + 328. i)T + (1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (350. - 607. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (49.5 - 28.5i)T + (1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + (-267. - 154. i)T + (1.78e5 + 3.09e5i)T^{2} \)
73 \( 1 - 389. iT - 3.89e5T^{2} \)
79 \( 1 - 901.T + 4.93e5T^{2} \)
83 \( 1 + 687. iT - 5.71e5T^{2} \)
89 \( 1 + (-927. + 535. i)T + (3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + (1.51e3 + 877. i)T + (4.56e5 + 7.90e5i)T^{2} \)
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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

−13.20577273449940885582999036497, −11.93138198436358746994662213739, −10.62949163342451828854632225461, −9.741474704147710934502419325378, −8.844316081680791063959505050449, −7.65262821125048351548745010225, −6.85983210228863351597068601705, −5.55774146400745512084857634167, −3.38575986463969649093693677978, −0.881693975055655711825576983157, 1.08648428413736721756562809407, 2.70912744466107263606092106516, 4.67947398457206658774981201108, 6.56357460656310603197347596007, 7.84799364439858443385826512311, 9.131305719330767331360972877237, 9.538500867301798025514737542315, 10.76088981496111298117279945107, 11.78165489959414372743459058855, 12.49775994281198609275225026326

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