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Properties

Label 2-108-27.22-c1-0-1
Degree $2$
Conductor $108$
Sign $0.541 + 0.840i$
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  + (−1.54 − 0.782i)3-s + (2.29 − 0.836i)5-s + (−0.775 − 4.39i)7-s + (1.77 + 2.41i)9-s + (2.73 + 0.996i)11-s + (−2.01 − 1.69i)13-s + (−4.20 − 0.505i)15-s + (−1.67 + 2.89i)17-s + (1.02 + 1.77i)19-s + (−2.24 + 7.40i)21-s + (−1.60 + 9.11i)23-s + (0.754 − 0.632i)25-s + (−0.851 − 5.12i)27-s + (5.30 − 4.45i)29-s + (−0.380 + 2.15i)31-s + ⋯
L(s)  = 1  + (−0.892 − 0.451i)3-s + (1.02 − 0.374i)5-s + (−0.293 − 1.66i)7-s + (0.591 + 0.806i)9-s + (0.825 + 0.300i)11-s + (−0.559 − 0.469i)13-s + (−1.08 − 0.130i)15-s + (−0.405 + 0.701i)17-s + (0.234 + 0.406i)19-s + (−0.489 + 1.61i)21-s + (−0.335 + 1.90i)23-s + (0.150 − 0.126i)25-s + (−0.163 − 0.986i)27-s + (0.985 − 0.826i)29-s + (−0.0683 + 0.387i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.541 + 0.840i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.541 + 0.840i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(108\)    =    \(2^{2} \cdot 3^{3}\)
Sign: $0.541 + 0.840i$
Analytic conductor: \(0.862384\)
Root analytic conductor: \(0.928646\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{108} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 108,\ (\ :1/2),\ 0.541 + 0.840i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.795395 - 0.434016i\)
\(L(\frac12)\) \(\approx\) \(0.795395 - 0.434016i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (1.54 + 0.782i)T \)
good5 \( 1 + (-2.29 + 0.836i)T + (3.83 - 3.21i)T^{2} \)
7 \( 1 + (0.775 + 4.39i)T + (-6.57 + 2.39i)T^{2} \)
11 \( 1 + (-2.73 - 0.996i)T + (8.42 + 7.07i)T^{2} \)
13 \( 1 + (2.01 + 1.69i)T + (2.25 + 12.8i)T^{2} \)
17 \( 1 + (1.67 - 2.89i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.02 - 1.77i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.60 - 9.11i)T + (-21.6 - 7.86i)T^{2} \)
29 \( 1 + (-5.30 + 4.45i)T + (5.03 - 28.5i)T^{2} \)
31 \( 1 + (0.380 - 2.15i)T + (-29.1 - 10.6i)T^{2} \)
37 \( 1 + (-0.708 + 1.22i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-3.13 - 2.63i)T + (7.11 + 40.3i)T^{2} \)
43 \( 1 + (-4.42 - 1.61i)T + (32.9 + 27.6i)T^{2} \)
47 \( 1 + (1.03 + 5.89i)T + (-44.1 + 16.0i)T^{2} \)
53 \( 1 - 1.97T + 53T^{2} \)
59 \( 1 + (6.20 - 2.25i)T + (45.1 - 37.9i)T^{2} \)
61 \( 1 + (1.25 + 7.11i)T + (-57.3 + 20.8i)T^{2} \)
67 \( 1 + (-2.37 - 1.99i)T + (11.6 + 65.9i)T^{2} \)
71 \( 1 + (6.60 - 11.4i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-6.40 - 11.0i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-1.57 + 1.32i)T + (13.7 - 77.7i)T^{2} \)
83 \( 1 + (1.20 - 1.00i)T + (14.4 - 81.7i)T^{2} \)
89 \( 1 + (6.88 + 11.9i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (13.9 + 5.07i)T + (74.3 + 62.3i)T^{2} \)
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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

−13.45680352324409844342954896705, −12.67004416543362994625764323842, −11.45247380088344718307282772414, −10.26596116777743875815886781018, −9.672848669105460233247275461517, −7.72957661745890165019779170425, −6.72373874156135852898162938676, −5.62778089009284809372867831713, −4.17191447136430877396781881841, −1.40216265067473815829546796609, 2.54560240619438872131859151210, 4.77494682327405173082418574989, 6.00224963144322058079598046451, 6.62585587020307680406713324687, 8.956305149505249197869943582397, 9.534314203738596936044921970865, 10.72176699498075644148634772598, 11.89622216057005530225982915097, 12.49005446328494658280167166316, 14.01185836826910707411121795933

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