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Properties

Label 2-108-36.11-c1-0-0
Degree $2$
Conductor $108$
Sign $0.208 - 0.978i$
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  + (0.774 + 1.18i)2-s + (−0.801 + 1.83i)4-s + (−0.686 + 0.396i)5-s + (2.35 + 1.35i)7-s + (−2.78 + 0.469i)8-s + (−1 − 0.505i)10-s + (1.71 − 2.96i)11-s + (−1.68 − 2.92i)13-s + (0.213 + 3.83i)14-s + (−2.71 − 2.93i)16-s − 2.52i·17-s − 2.20i·19-s + (−0.175 − 1.57i)20-s + (4.83 − 0.269i)22-s + (1.07 + 1.86i)23-s + ⋯
L(s)  = 1  + (0.547 + 0.836i)2-s + (−0.400 + 0.916i)4-s + (−0.306 + 0.177i)5-s + (0.888 + 0.513i)7-s + (−0.986 + 0.166i)8-s + (−0.316 − 0.159i)10-s + (0.516 − 0.894i)11-s + (−0.467 − 0.809i)13-s + (0.0570 + 1.02i)14-s + (−0.678 − 0.734i)16-s − 0.612i·17-s − 0.506i·19-s + (−0.0393 − 0.352i)20-s + (1.03 − 0.0574i)22-s + (0.224 + 0.388i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.208 - 0.978i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.208 - 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(108\)    =    \(2^{2} \cdot 3^{3}\)
Sign: $0.208 - 0.978i$
Analytic conductor: \(0.862384\)
Root analytic conductor: \(0.928646\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{108} (35, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 108,\ (\ :1/2),\ 0.208 - 0.978i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.991492 + 0.802692i\)
\(L(\frac12)\) \(\approx\) \(0.991492 + 0.802692i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.774 - 1.18i)T \)
3 \( 1 \)
good5 \( 1 + (0.686 - 0.396i)T + (2.5 - 4.33i)T^{2} \)
7 \( 1 + (-2.35 - 1.35i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-1.71 + 2.96i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (1.68 + 2.92i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + 2.52iT - 17T^{2} \)
19 \( 1 + 2.20iT - 19T^{2} \)
23 \( 1 + (-1.07 - 1.86i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.686 - 0.396i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (1.47 - 0.852i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 - 4.74T + 37T^{2} \)
41 \( 1 + (0.127 - 0.0737i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (6.01 + 3.47i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (5.77 - 10.0i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 8.51iT - 53T^{2} \)
59 \( 1 + (2.58 + 4.48i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.68 - 2.92i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (6.01 - 3.47i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 1.75T + 71T^{2} \)
73 \( 1 + 2.37T + 73T^{2} \)
79 \( 1 + (-8.80 - 5.08i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-3.62 + 6.28i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 5.34iT - 89T^{2} \)
97 \( 1 + (-6.24 + 10.8i)T + (-48.5 - 84.0i)T^{2} \)
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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

−14.11549235287910860712219332391, −13.10688876830430578272266023626, −11.89508451578533425119532007349, −11.16731711362629497075761526678, −9.301510483207408222556147131753, −8.246253221894802420967215104283, −7.30904186905061507455959438970, −5.86643302778798293221600635166, −4.80397396391423472332909222377, −3.16107461221181382064433288203, 1.83737160902955030942261527507, 4.00679243161964543249884286083, 4.85673845474614830510142843406, 6.58335140505774027686789856750, 8.096312639468680913296560910260, 9.478091681584887851899897100131, 10.48458498560501905858097823936, 11.60691497542532388159915836910, 12.24680014391773285566333116619, 13.41268934059620128606386950314

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