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Properties

Label 2-117-117.29-c2-0-23
Degree $2$
Conductor $117$
Sign $-0.999 + 0.00692i$
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  − 1.67i·2-s + (0.0978 − 2.99i)3-s + 1.20·4-s + (−8.08 + 4.67i)5-s + (−5.00 − 0.163i)6-s + (−4.17 − 7.23i)7-s − 8.70i·8-s + (−8.98 − 0.586i)9-s + (7.80 + 13.5i)10-s + 2.35i·11-s + (0.118 − 3.62i)12-s + (6.74 − 11.1i)13-s + (−12.0 + 6.98i)14-s + (13.2 + 24.7i)15-s − 9.70·16-s + (5.30 + 3.06i)17-s + ⋯
L(s)  = 1  − 0.835i·2-s + (0.0326 − 0.999i)3-s + 0.302·4-s + (−1.61 + 0.934i)5-s + (−0.834 − 0.0272i)6-s + (−0.596 − 1.03i)7-s − 1.08i·8-s + (−0.997 − 0.0651i)9-s + (0.780 + 1.35i)10-s + 0.214i·11-s + (0.00985 − 0.302i)12-s + (0.518 − 0.855i)13-s + (−0.863 + 0.498i)14-s + (0.880 + 1.64i)15-s − 0.606·16-s + (0.311 + 0.180i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.00692i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.999 + 0.00692i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(117\)    =    \(3^{2} \cdot 13\)
Sign: $-0.999 + 0.00692i$
Analytic conductor: \(3.18801\)
Root analytic conductor: \(1.78550\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{117} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 117,\ (\ :1),\ -0.999 + 0.00692i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.00319982 - 0.924053i\)
\(L(\frac12)\) \(\approx\) \(0.00319982 - 0.924053i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.0978 + 2.99i)T \)
13 \( 1 + (-6.74 + 11.1i)T \)
good2 \( 1 + 1.67iT - 4T^{2} \)
5 \( 1 + (8.08 - 4.67i)T + (12.5 - 21.6i)T^{2} \)
7 \( 1 + (4.17 + 7.23i)T + (-24.5 + 42.4i)T^{2} \)
11 \( 1 - 2.35iT - 121T^{2} \)
17 \( 1 + (-5.30 - 3.06i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (-3.19 + 5.52i)T + (-180.5 - 312. i)T^{2} \)
23 \( 1 + (-18.1 - 10.4i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + 32.8iT - 841T^{2} \)
31 \( 1 + (10.3 + 17.8i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 + (-14.1 - 24.4i)T + (-684.5 + 1.18e3i)T^{2} \)
41 \( 1 + (23.6 + 13.6i)T + (840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (-37.1 - 64.3i)T + (-924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (20.4 + 11.7i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + 97.2iT - 2.80e3T^{2} \)
59 \( 1 + 8.91iT - 3.48e3T^{2} \)
61 \( 1 + (-8.35 - 14.4i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-2.20 + 3.82i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + (-12.9 - 7.48i)T + (2.52e3 + 4.36e3i)T^{2} \)
73 \( 1 + 42.4T + 5.32e3T^{2} \)
79 \( 1 + (44.1 - 76.4i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (-55.1 - 31.8i)T + (3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 + (72.0 - 41.6i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + (-55.6 - 96.4i)T + (-4.70e3 + 8.14e3i)T^{2} \)
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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

−12.65471493722357400780001994130, −11.53722072656795103767938110639, −11.10611308968794301855906478192, −10.03949086187265217365444397286, −8.011714534097058948595344686486, −7.28332320178990592117531580332, −6.48261732178731370649838418197, −3.77890199743493460950128782825, −2.94782743252884082236048289527, −0.65397975460085962594303615888, 3.29162005460365956684467515672, 4.71477059031801988228564370847, 5.81665246088747539162283735739, 7.32725485543480719676437978851, 8.674781825851207468922780637737, 8.954591104519073443644731789797, 10.89445973061620541141072701500, 11.73462420926268263480801215885, 12.48970332251853060414567413904, 14.29504590184271147060820264291

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