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Properties

Label 2-117-117.29-c2-0-22
Degree $2$
Conductor $117$
Sign $-0.998 + 0.0488i$
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  − 2.75i·2-s + (−1.69 − 2.47i)3-s − 3.61·4-s + (7.95 − 4.59i)5-s + (−6.82 + 4.68i)6-s + (−2.71 − 4.71i)7-s − 1.05i·8-s + (−3.23 + 8.39i)9-s + (−12.6 − 21.9i)10-s + 18.2i·11-s + (6.14 + 8.94i)12-s + (12.7 + 2.59i)13-s + (−13.0 + 7.50i)14-s + (−24.8 − 11.8i)15-s − 17.3·16-s + (3.53 + 2.04i)17-s + ⋯
L(s)  = 1  − 1.37i·2-s + (−0.565 − 0.824i)3-s − 0.904·4-s + (1.59 − 0.918i)5-s + (−1.13 + 0.780i)6-s + (−0.388 − 0.672i)7-s − 0.132i·8-s + (−0.359 + 0.933i)9-s + (−1.26 − 2.19i)10-s + 1.66i·11-s + (0.511 + 0.745i)12-s + (0.979 + 0.199i)13-s + (−0.928 + 0.536i)14-s + (−1.65 − 0.791i)15-s − 1.08·16-s + (0.208 + 0.120i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 + 0.0488i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.998 + 0.0488i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(117\)    =    \(3^{2} \cdot 13\)
Sign: $-0.998 + 0.0488i$
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: \((2,\ 117,\ (\ :1),\ -0.998 + 0.0488i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.0342655 - 1.40129i\)
\(L(\frac12)\) \(\approx\) \(0.0342655 - 1.40129i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.69 + 2.47i)T \)
13 \( 1 + (-12.7 - 2.59i)T \)
good2 \( 1 + 2.75iT - 4T^{2} \)
5 \( 1 + (-7.95 + 4.59i)T + (12.5 - 21.6i)T^{2} \)
7 \( 1 + (2.71 + 4.71i)T + (-24.5 + 42.4i)T^{2} \)
11 \( 1 - 18.2iT - 121T^{2} \)
17 \( 1 + (-3.53 - 2.04i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (4.68 - 8.11i)T + (-180.5 - 312. i)T^{2} \)
23 \( 1 + (2.18 + 1.26i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + 10.3iT - 841T^{2} \)
31 \( 1 + (-7.76 - 13.4i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 + (23.6 + 41.0i)T + (-684.5 + 1.18e3i)T^{2} \)
41 \( 1 + (-34.8 - 20.1i)T + (840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (22.9 + 39.8i)T + (-924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (-19.2 - 11.1i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 - 73.5iT - 2.80e3T^{2} \)
59 \( 1 + 3.38iT - 3.48e3T^{2} \)
61 \( 1 + (-17.2 - 29.9i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (37.9 - 65.7i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + (-8.47 - 4.89i)T + (2.52e3 + 4.36e3i)T^{2} \)
73 \( 1 + 90.9T + 5.32e3T^{2} \)
79 \( 1 + (7.46 - 12.9i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (37.6 + 21.7i)T + (3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 + (-71.5 + 41.3i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + (46.4 + 80.4i)T + (-4.70e3 + 8.14e3i)T^{2} \)
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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

−12.73540550423292872363048868809, −12.02154373851907887340719381174, −10.55678079120905461759159040336, −10.01754996238781009805571275737, −8.918133532732633451415068240819, −7.05579147864029606350891047057, −5.88161831909306554081344685921, −4.42054636727477143742406612847, −2.14530398241482855097306282301, −1.22639052373320478898290845458, 3.06717281110714732958372549269, 5.38455388666814728094105976478, 6.03855822340821056584620887440, 6.54873653525190003366095676929, 8.549401381034314785005661440551, 9.379328778559640208696627767770, 10.59048792790288128758674503255, 11.38499175540091103975124622554, 13.32534244139030857166578233379, 14.05534778413252816378373406259

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