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Properties

Label 2-117-117.43-c3-0-2
Degree $2$
Conductor $117$
Sign $-0.741 + 0.671i$
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  + (−4.65 + 2.68i)2-s + (1.15 + 5.06i)3-s + (10.4 − 18.0i)4-s + (10.1 − 5.86i)5-s + (−18.9 − 20.4i)6-s − 8.42i·7-s + 68.9i·8-s + (−24.3 + 11.7i)9-s + (−31.5 + 54.5i)10-s + (−55.4 + 32.0i)11-s + (103. + 31.8i)12-s + (−38.7 − 26.4i)13-s + (22.6 + 39.1i)14-s + (41.4 + 44.7i)15-s + (−101. − 176. i)16-s + (30.6 + 53.0i)17-s + ⋯
L(s)  = 1  + (−1.64 + 0.949i)2-s + (0.222 + 0.974i)3-s + (1.30 − 2.25i)4-s + (0.909 − 0.524i)5-s + (−1.29 − 1.39i)6-s − 0.454i·7-s + 3.04i·8-s + (−0.900 + 0.434i)9-s + (−0.996 + 1.72i)10-s + (−1.52 + 0.877i)11-s + (2.48 + 0.767i)12-s + (−0.826 − 0.563i)13-s + (0.431 + 0.747i)14-s + (0.714 + 0.769i)15-s + (−1.58 − 2.75i)16-s + (0.436 + 0.756i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.741 + 0.671i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.741 + 0.671i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(117\)    =    \(3^{2} \cdot 13\)
Sign: $-0.741 + 0.671i$
Analytic conductor: \(6.90322\)
Root analytic conductor: \(2.62739\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{117} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 117,\ (\ :3/2),\ -0.741 + 0.671i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.101113 - 0.262192i\)
\(L(\frac12)\) \(\approx\) \(0.101113 - 0.262192i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.15 - 5.06i)T \)
13 \( 1 + (38.7 + 26.4i)T \)
good2 \( 1 + (4.65 - 2.68i)T + (4 - 6.92i)T^{2} \)
5 \( 1 + (-10.1 + 5.86i)T + (62.5 - 108. i)T^{2} \)
7 \( 1 + 8.42iT - 343T^{2} \)
11 \( 1 + (55.4 - 32.0i)T + (665.5 - 1.15e3i)T^{2} \)
17 \( 1 + (-30.6 - 53.0i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (86.4 - 49.9i)T + (3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + 106.T + 1.21e4T^{2} \)
29 \( 1 + (-72.3 - 125. i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (94.2 - 54.3i)T + (1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (4.75 + 2.74i)T + (2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 - 66.6iT - 6.89e4T^{2} \)
43 \( 1 - 331.T + 7.95e4T^{2} \)
47 \( 1 + (181. + 104. i)T + (5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + 303.T + 1.48e5T^{2} \)
59 \( 1 + (576. + 332. i)T + (1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 - 636.T + 2.26e5T^{2} \)
67 \( 1 - 138. iT - 3.00e5T^{2} \)
71 \( 1 + (303. - 175. i)T + (1.78e5 - 3.09e5i)T^{2} \)
73 \( 1 + 497. iT - 3.89e5T^{2} \)
79 \( 1 + (1.08 - 1.87i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (-122. - 70.8i)T + (2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 + (71.9 + 41.5i)T + (3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 - 510. iT - 9.12e5T^{2} \)
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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

−14.24837799625505357806226848259, −12.70953000800045201468239199780, −10.59654234396214791272125831006, −10.25698743382883838839332583036, −9.528330716970586761839253574481, −8.368244534217071663748407993227, −7.58197968408878284870094761615, −5.91551836310522877792466778480, −5.00551599057266802427988604548, −2.08617030336363324387045663764, 0.22156887258503452094864377313, 2.21425833975579554618019564897, 2.71808599369814176031541099052, 6.05329047659187521770835720503, 7.34012588180400346186298325532, 8.225969374069199945754255820577, 9.253728250899668658568991242054, 10.22127629977443927855227194348, 11.19009276284356258798431461387, 12.16700655449680308005447550270

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