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Properties

Label 2-108-12.11-c3-0-2
Degree $2$
Conductor $108$
Sign $-0.955 + 0.293i$
Analytic cond. $6.37220$
Root an. cond. $2.52432$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.419 + 2.79i)2-s + (−7.64 + 2.34i)4-s − 1.49i·5-s + 26.1i·7-s + (−9.78 − 20.4i)8-s + (4.18 − 0.627i)10-s − 56.3·11-s − 41.3·13-s + (−73.2 + 10.9i)14-s + (52.9 − 35.9i)16-s − 51.0i·17-s + 79.0i·19-s + (3.51 + 11.4i)20-s + (−23.6 − 157. i)22-s − 27.3·23-s + ⋯
L(s)  = 1  + (0.148 + 0.988i)2-s + (−0.955 + 0.293i)4-s − 0.133i·5-s + 1.41i·7-s + (−0.432 − 0.901i)8-s + (0.132 − 0.0198i)10-s − 1.54·11-s − 0.881·13-s + (−1.39 + 0.209i)14-s + (0.827 − 0.561i)16-s − 0.728i·17-s + 0.954i·19-s + (0.0392 + 0.127i)20-s + (−0.229 − 1.52i)22-s − 0.248·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.955 + 0.293i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.955 + 0.293i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(108\)    =    \(2^{2} \cdot 3^{3}\)
Sign: $-0.955 + 0.293i$
Analytic conductor: \(6.37220\)
Root analytic conductor: \(2.52432\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{108} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 108,\ (\ :3/2),\ -0.955 + 0.293i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.117878 - 0.785220i\)
\(L(\frac12)\) \(\approx\) \(0.117878 - 0.785220i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.419 - 2.79i)T \)
3 \( 1 \)
good5 \( 1 + 1.49iT - 125T^{2} \)
7 \( 1 - 26.1iT - 343T^{2} \)
11 \( 1 + 56.3T + 1.33e3T^{2} \)
13 \( 1 + 41.3T + 2.19e3T^{2} \)
17 \( 1 + 51.0iT - 4.91e3T^{2} \)
19 \( 1 - 79.0iT - 6.85e3T^{2} \)
23 \( 1 + 27.3T + 1.21e4T^{2} \)
29 \( 1 - 134. iT - 2.43e4T^{2} \)
31 \( 1 - 187. iT - 2.97e4T^{2} \)
37 \( 1 + 196.T + 5.06e4T^{2} \)
41 \( 1 + 298. iT - 6.89e4T^{2} \)
43 \( 1 - 465. iT - 7.95e4T^{2} \)
47 \( 1 - 373.T + 1.03e5T^{2} \)
53 \( 1 - 620. iT - 1.48e5T^{2} \)
59 \( 1 - 321.T + 2.05e5T^{2} \)
61 \( 1 - 674.T + 2.26e5T^{2} \)
67 \( 1 + 576. iT - 3.00e5T^{2} \)
71 \( 1 + 223.T + 3.57e5T^{2} \)
73 \( 1 - 70.1T + 3.89e5T^{2} \)
79 \( 1 + 1.05e3iT - 4.93e5T^{2} \)
83 \( 1 + 1.21e3T + 5.71e5T^{2} \)
89 \( 1 - 1.34e3iT - 7.04e5T^{2} \)
97 \( 1 + 576.T + 9.12e5T^{2} \)
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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

−14.00901541418725723207767473108, −12.67396630576749896379312081209, −12.22020383739649237705364211551, −10.39989383955569810216331659017, −9.188287060584829001047179541825, −8.273063835946442654240795925046, −7.16788307045704653503900155215, −5.63380386291959675188505703991, −4.97360446399956787010489099854, −2.82343097697133271625410694667, 0.40363863507521334178840110270, 2.51266461732731790883521016926, 4.07148051659690706220469174416, 5.25765727952813665531752714032, 7.17230752786748737743345569730, 8.336995056567058401832769992642, 9.953928918539182458067417306933, 10.47127404205020536756786962905, 11.44062131979484518613368177852, 12.84918884044671511239894361725

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