Properties

Label 2-108-108.47-c1-0-10
Degree $2$
Conductor $108$
Sign $0.297 + 0.954i$
Analytic cond. $0.862384$
Root an. cond. $0.928646$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.20 − 0.747i)2-s + (0.644 − 1.60i)3-s + (0.881 + 1.79i)4-s + (3.32 − 0.585i)5-s + (−1.97 + 1.44i)6-s + (−1.72 + 2.05i)7-s + (0.285 − 2.81i)8-s + (−2.16 − 2.07i)9-s + (−4.42 − 1.78i)10-s + (0.506 − 2.87i)11-s + (3.45 − 0.259i)12-s + (0.552 + 0.201i)13-s + (3.61 − 1.17i)14-s + (1.19 − 5.71i)15-s + (−2.44 + 3.16i)16-s + (−6.36 + 3.67i)17-s + ⋯
L(s)  = 1  + (−0.848 − 0.528i)2-s + (0.372 − 0.928i)3-s + (0.440 + 0.897i)4-s + (1.48 − 0.262i)5-s + (−0.806 + 0.591i)6-s + (−0.652 + 0.777i)7-s + (0.100 − 0.994i)8-s + (−0.723 − 0.690i)9-s + (−1.39 − 0.563i)10-s + (0.152 − 0.866i)11-s + (0.997 − 0.0749i)12-s + (0.153 + 0.0557i)13-s + (0.965 − 0.314i)14-s + (0.309 − 1.47i)15-s + (−0.611 + 0.791i)16-s + (−1.54 + 0.890i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.297 + 0.954i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.297 + 0.954i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(108\)    =    \(2^{2} \cdot 3^{3}\)
Sign: $0.297 + 0.954i$
Analytic conductor: \(0.862384\)
Root analytic conductor: \(0.928646\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{108} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 108,\ (\ :1/2),\ 0.297 + 0.954i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.722650 - 0.531934i\)
\(L(\frac12)\) \(\approx\) \(0.722650 - 0.531934i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.20 + 0.747i)T \)
3 \( 1 + (-0.644 + 1.60i)T \)
good5 \( 1 + (-3.32 + 0.585i)T + (4.69 - 1.71i)T^{2} \)
7 \( 1 + (1.72 - 2.05i)T + (-1.21 - 6.89i)T^{2} \)
11 \( 1 + (-0.506 + 2.87i)T + (-10.3 - 3.76i)T^{2} \)
13 \( 1 + (-0.552 - 0.201i)T + (9.95 + 8.35i)T^{2} \)
17 \( 1 + (6.36 - 3.67i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.82 - 1.63i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.365 + 0.306i)T + (3.99 - 22.6i)T^{2} \)
29 \( 1 + (-1.49 - 4.10i)T + (-22.2 + 18.6i)T^{2} \)
31 \( 1 + (-3.73 - 4.45i)T + (-5.38 + 30.5i)T^{2} \)
37 \( 1 + (1.59 + 2.76i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (1.09 - 3.01i)T + (-31.4 - 26.3i)T^{2} \)
43 \( 1 + (1.79 + 0.316i)T + (40.4 + 14.7i)T^{2} \)
47 \( 1 + (0.940 + 0.789i)T + (8.16 + 46.2i)T^{2} \)
53 \( 1 + 7.37iT - 53T^{2} \)
59 \( 1 + (0.718 + 4.07i)T + (-55.4 + 20.1i)T^{2} \)
61 \( 1 + (-0.421 - 0.353i)T + (10.5 + 60.0i)T^{2} \)
67 \( 1 + (3.46 - 9.51i)T + (-51.3 - 43.0i)T^{2} \)
71 \( 1 + (0.616 + 1.06i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-1.18 + 2.04i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (3.33 + 9.16i)T + (-60.5 + 50.7i)T^{2} \)
83 \( 1 + (16.8 - 6.15i)T + (63.5 - 53.3i)T^{2} \)
89 \( 1 + (4.81 + 2.77i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-1.83 + 10.4i)T + (-91.1 - 33.1i)T^{2} \)
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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

−13.19216985660394994615131022092, −12.63831820035056113313409489471, −11.42021539929533368830173758108, −10.10106613242468224227567721450, −8.980091072198626124229688542788, −8.555691836124369607825945941104, −6.73130026014049292978152716448, −5.94346068934688029368477542518, −3.00472154309855540433749899981, −1.72679897074524766963755363105, 2.45366194142412557905455544177, 4.75703880877912437108946959914, 6.17628327937766167887479047605, 7.22426556618135830651140410537, 8.913552983419003000691775197901, 9.737109437443052936008854188805, 10.16004911535749931050589140326, 11.28725143694624612171661760182, 13.52457880843652458974865284119, 13.88634890493993509366353777696

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