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Properties

Label 2-117-39.17-c2-0-2
Degree $2$
Conductor $117$
Sign $0.324 - 0.945i$
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  + (−0.718 + 1.24i)2-s + (0.967 + 1.67i)4-s + 3.30·5-s + (10.5 − 6.06i)7-s − 8.52·8-s + (−2.37 + 4.11i)10-s + (−7.35 + 12.7i)11-s + (3.48 + 12.5i)13-s + 17.4i·14-s + (2.25 − 3.90i)16-s + (19.9 − 11.4i)17-s + (−9.63 + 5.56i)19-s + (3.19 + 5.54i)20-s + (−10.5 − 18.3i)22-s + (3.03 + 1.75i)23-s + ⋯
L(s)  = 1  + (−0.359 + 0.622i)2-s + (0.241 + 0.419i)4-s + 0.661·5-s + (1.50 − 0.866i)7-s − 1.06·8-s + (−0.237 + 0.411i)10-s + (−0.668 + 1.15i)11-s + (0.268 + 0.963i)13-s + 1.24i·14-s + (0.141 − 0.244i)16-s + (1.17 − 0.676i)17-s + (−0.506 + 0.292i)19-s + (0.159 + 0.277i)20-s + (−0.480 − 0.831i)22-s + (0.131 + 0.0761i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.324 - 0.945i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.324 - 0.945i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(117\)    =    \(3^{2} \cdot 13\)
Sign: $0.324 - 0.945i$
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: \((2,\ 117,\ (\ :1),\ 0.324 - 0.945i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.17713 + 0.840248i\)
\(L(\frac12)\) \(\approx\) \(1.17713 + 0.840248i\)
\(L(2)\) 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 \( 1 + (-3.48 - 12.5i)T \)
good2 \( 1 + (0.718 - 1.24i)T + (-2 - 3.46i)T^{2} \)
5 \( 1 - 3.30T + 25T^{2} \)
7 \( 1 + (-10.5 + 6.06i)T + (24.5 - 42.4i)T^{2} \)
11 \( 1 + (7.35 - 12.7i)T + (-60.5 - 104. i)T^{2} \)
17 \( 1 + (-19.9 + 11.4i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (9.63 - 5.56i)T + (180.5 - 312. i)T^{2} \)
23 \( 1 + (-3.03 - 1.75i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + (10.8 + 6.23i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + 29.9iT - 961T^{2} \)
37 \( 1 + (22.5 + 13.0i)T + (684.5 + 1.18e3i)T^{2} \)
41 \( 1 + (-37.9 + 65.8i)T + (-840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (-4.34 - 7.53i)T + (-924.5 + 1.60e3i)T^{2} \)
47 \( 1 - 31.3T + 2.20e3T^{2} \)
53 \( 1 - 52.3iT - 2.80e3T^{2} \)
59 \( 1 + (20.6 + 35.6i)T + (-1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-25.1 - 43.5i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (102. + 59.3i)T + (2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + (16.8 + 29.1i)T + (-2.52e3 + 4.36e3i)T^{2} \)
73 \( 1 - 5.21iT - 5.32e3T^{2} \)
79 \( 1 + 39.7T + 6.24e3T^{2} \)
83 \( 1 + 141.T + 6.88e3T^{2} \)
89 \( 1 + (-11.9 + 20.7i)T + (-3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + (-43.0 + 24.8i)T + (4.70e3 - 8.14e3i)T^{2} \)
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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

−13.75145580640151293661189416901, −12.39882946742280462640385673609, −11.42547602190245696011863112783, −10.26429865561453145507132254915, −9.088072462717220562211924655302, −7.73198848019027849280887473022, −7.29031724666143410762043729178, −5.69109280506249585804412492478, −4.25889349466369117048039222329, −2.01410289120608439050958902970, 1.43868524602874116009912958435, 2.85959878445791617715869984083, 5.36687373681717188278096749195, 5.91036910376092960852970102362, 8.035493524072535014172082745962, 8.813709678232729190204876469786, 10.20813169162465585369224981793, 10.90942593561793081400085306334, 11.77454437543789409960041862591, 12.93362456762354824144165408348

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