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Properties

Label 2-117-117.103-c3-0-14
Degree $2$
Conductor $117$
Sign $-0.893 - 0.449i$
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  + (−0.457 + 0.263i)2-s + (0.870 + 5.12i)3-s + (−3.86 + 6.68i)4-s + (14.6 + 8.47i)5-s + (−1.74 − 2.11i)6-s + (−18.0 + 10.4i)7-s − 8.29i·8-s + (−25.4 + 8.91i)9-s − 8.94·10-s + (37.8 − 21.8i)11-s + (−37.6 − 13.9i)12-s + (38.8 − 26.2i)13-s + (5.51 − 9.55i)14-s + (−30.6 + 82.5i)15-s + (−28.6 − 49.7i)16-s − 84.5·17-s + ⋯
L(s)  = 1  + (−0.161 + 0.0933i)2-s + (0.167 + 0.985i)3-s + (−0.482 + 0.835i)4-s + (1.31 + 0.757i)5-s + (−0.119 − 0.143i)6-s + (−0.977 + 0.564i)7-s − 0.366i·8-s + (−0.943 + 0.330i)9-s − 0.282·10-s + (1.03 − 0.598i)11-s + (−0.904 − 0.335i)12-s + (0.828 − 0.559i)13-s + (0.105 − 0.182i)14-s + (−0.527 + 1.42i)15-s + (−0.448 − 0.776i)16-s − 1.20·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.893 - 0.449i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.893 - 0.449i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(117\)    =    \(3^{2} \cdot 13\)
Sign: $-0.893 - 0.449i$
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: \((2,\ 117,\ (\ :3/2),\ -0.893 - 0.449i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.328898 + 1.38645i\)
\(L(\frac12)\) \(\approx\) \(0.328898 + 1.38645i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.870 - 5.12i)T \)
13 \( 1 + (-38.8 + 26.2i)T \)
good2 \( 1 + (0.457 - 0.263i)T + (4 - 6.92i)T^{2} \)
5 \( 1 + (-14.6 - 8.47i)T + (62.5 + 108. i)T^{2} \)
7 \( 1 + (18.0 - 10.4i)T + (171.5 - 297. i)T^{2} \)
11 \( 1 + (-37.8 + 21.8i)T + (665.5 - 1.15e3i)T^{2} \)
17 \( 1 + 84.5T + 4.91e3T^{2} \)
19 \( 1 - 48.1iT - 6.85e3T^{2} \)
23 \( 1 + (38.9 - 67.4i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (-105. - 182. i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (-144. - 83.4i)T + (1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + 379. iT - 5.06e4T^{2} \)
41 \( 1 + (175. + 101. i)T + (3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (-189. - 329. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (-306. + 177. i)T + (5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + 179.T + 1.48e5T^{2} \)
59 \( 1 + (-273. - 157. i)T + (1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (133. + 231. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-400. - 231. i)T + (1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 492. iT - 3.57e5T^{2} \)
73 \( 1 - 483. iT - 3.89e5T^{2} \)
79 \( 1 + (41.1 + 71.1i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (749. - 432. i)T + (2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 - 1.06e3iT - 7.04e5T^{2} \)
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

−13.70599120660455874546388184696, −12.62062647157117258440046372692, −11.18919889577209862908443969958, −10.10167391167355170674475241567, −9.218752323550289333876169001254, −8.611020159029978244243391374305, −6.62123928208031165987642517770, −5.70260657005273084359934246013, −3.81438144458779907408980505447, −2.81341590786741913177244874145, 0.820620951744775713348412732479, 2.03715937445336839066246304978, 4.49596360296510368202539770354, 6.24283004084043792418879922141, 6.52880016391080081196803901877, 8.676710078104662859835049269027, 9.317662028764823630588479779055, 10.20356570830029254289126933225, 11.70100618744831439303442855191, 13.04117553083482286798433896378

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