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Properties

Label 2-117-39.17-c2-0-1
Degree $2$
Conductor $117$
Sign $-0.792 - 0.609i$
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  + (−0.201 + 0.349i)2-s + (1.91 + 3.32i)4-s − 6.09·5-s + (−9.03 + 5.21i)7-s − 3.16·8-s + (1.22 − 2.12i)10-s + (1.06 − 1.83i)11-s + (12.9 + 0.692i)13-s − 4.21i·14-s + (−7.03 + 12.1i)16-s + (−26.4 + 15.2i)17-s + (3.26 − 1.88i)19-s + (−11.6 − 20.2i)20-s + (0.428 + 0.741i)22-s + (28.7 + 16.6i)23-s + ⋯
L(s)  = 1  + (−0.100 + 0.174i)2-s + (0.479 + 0.830i)4-s − 1.21·5-s + (−1.29 + 0.745i)7-s − 0.395·8-s + (0.122 − 0.212i)10-s + (0.0964 − 0.167i)11-s + (0.998 + 0.0532i)13-s − 0.300i·14-s + (−0.439 + 0.761i)16-s + (−1.55 + 0.897i)17-s + (0.171 − 0.0992i)19-s + (−0.584 − 1.01i)20-s + (0.0194 + 0.0337i)22-s + (1.25 + 0.722i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.792 - 0.609i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.792 - 0.609i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(117\)    =    \(3^{2} \cdot 13\)
Sign: $-0.792 - 0.609i$
Analytic conductor: \(3.18801\)
Root analytic conductor: \(1.78550\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{117} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 117,\ (\ :1),\ -0.792 - 0.609i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.237725 + 0.698623i\)
\(L(\frac12)\) \(\approx\) \(0.237725 + 0.698623i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
13 \( 1 + (-12.9 - 0.692i)T \)
good2 \( 1 + (0.201 - 0.349i)T + (-2 - 3.46i)T^{2} \)
5 \( 1 + 6.09T + 25T^{2} \)
7 \( 1 + (9.03 - 5.21i)T + (24.5 - 42.4i)T^{2} \)
11 \( 1 + (-1.06 + 1.83i)T + (-60.5 - 104. i)T^{2} \)
17 \( 1 + (26.4 - 15.2i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (-3.26 + 1.88i)T + (180.5 - 312. i)T^{2} \)
23 \( 1 + (-28.7 - 16.6i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + (-35.3 - 20.4i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + 21.3iT - 961T^{2} \)
37 \( 1 + (-4.57 - 2.64i)T + (684.5 + 1.18e3i)T^{2} \)
41 \( 1 + (-12.4 + 21.6i)T + (-840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (0.388 + 0.673i)T + (-924.5 + 1.60e3i)T^{2} \)
47 \( 1 + 29.1T + 2.20e3T^{2} \)
53 \( 1 - 51.3iT - 2.80e3T^{2} \)
59 \( 1 + (-46.0 - 79.7i)T + (-1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (31.3 + 54.3i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-7.81 - 4.51i)T + (2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + (38.3 + 66.3i)T + (-2.52e3 + 4.36e3i)T^{2} \)
73 \( 1 - 124. iT - 5.32e3T^{2} \)
79 \( 1 - 0.898T + 6.24e3T^{2} \)
83 \( 1 + 97.0T + 6.88e3T^{2} \)
89 \( 1 + (8.09 - 14.0i)T + (-3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + (105. - 60.7i)T + (4.70e3 - 8.14e3i)T^{2} \)
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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

−13.31949874421345704013405444061, −12.64273803336084361453882823953, −11.67403674589327366640537763752, −10.89242043968348870371331500779, −9.057926203218429840487862690279, −8.399728792599686562807687417764, −7.07504324903172577713107445799, −6.19358572118670195791432460730, −4.00226100904252037948458340888, −2.97551198317308714586116462692, 0.52867536125554189063262702114, 3.08619383568959488511325257194, 4.52079737144504193092315420511, 6.44548920292800620766722815649, 7.06788496404960782339487716615, 8.682237381887945389640676090448, 9.855783240187869837447705060516, 10.90356676302431109027319727180, 11.58698874056072727830534860649, 12.87274315200476836035541756130

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