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Properties

Label 2-108-36.11-c1-0-1
Degree $2$
Conductor $108$
Sign $0.899 - 0.436i$
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  + (−1.02 + 0.977i)2-s + (0.0884 − 1.99i)4-s + (2.18 − 1.26i)5-s + (1.10 + 0.637i)7-s + (1.86 + 2.12i)8-s + (−1 + 3.42i)10-s + (−0.252 + 0.437i)11-s + (1.18 + 2.05i)13-s + (−1.75 + 0.428i)14-s + (−3.98 − 0.353i)16-s + 0.792i·17-s − 4.70i·19-s + (−2.32 − 4.47i)20-s + (−0.169 − 0.694i)22-s + (−1.61 − 2.78i)23-s + ⋯
L(s)  = 1  + (−0.722 + 0.691i)2-s + (0.0442 − 0.999i)4-s + (0.977 − 0.564i)5-s + (0.417 + 0.241i)7-s + (0.658 + 0.752i)8-s + (−0.316 + 1.08i)10-s + (−0.0761 + 0.131i)11-s + (0.328 + 0.569i)13-s + (−0.468 + 0.114i)14-s + (−0.996 − 0.0883i)16-s + 0.192i·17-s − 1.07i·19-s + (−0.520 − 1.00i)20-s + (−0.0361 − 0.148i)22-s + (−0.335 − 0.581i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.899 - 0.436i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.899 - 0.436i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(108\)    =    \(2^{2} \cdot 3^{3}\)
Sign: $0.899 - 0.436i$
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: \((2,\ 108,\ (\ :1/2),\ 0.899 - 0.436i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.840466 + 0.192918i\)
\(L(\frac12)\) \(\approx\) \(0.840466 + 0.192918i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.02 - 0.977i)T \)
3 \( 1 \)
good5 \( 1 + (-2.18 + 1.26i)T + (2.5 - 4.33i)T^{2} \)
7 \( 1 + (-1.10 - 0.637i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (0.252 - 0.437i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-1.18 - 2.05i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 - 0.792iT - 17T^{2} \)
19 \( 1 + 4.70iT - 19T^{2} \)
23 \( 1 + (1.61 + 2.78i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (2.18 + 1.26i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (7.04 - 4.06i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + 6.74T + 37T^{2} \)
41 \( 1 + (5.87 - 3.39i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-6.69 - 3.86i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (0.599 - 1.03i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 1.87iT - 53T^{2} \)
59 \( 1 + (-6.18 - 10.7i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.18 + 2.05i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-6.69 + 3.86i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 11.8T + 71T^{2} \)
73 \( 1 - 3.37T + 73T^{2} \)
79 \( 1 + (8.55 + 4.94i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-3.82 + 6.61i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 11.9iT - 89T^{2} \)
97 \( 1 + (5.24 - 9.08i)T + (-48.5 - 84.0i)T^{2} \)
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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

−13.94342525943148445893820859087, −12.99741641261377602716474940190, −11.47012266548341335782527206402, −10.34132661050047860723472875550, −9.234915357312198534067508075352, −8.584146592780211395204909406992, −7.12083477206621752187974704226, −5.89979243042656340630716085674, −4.84435683226521457078002092218, −1.82256491519157782820972525716, 1.93384698792762223064612800871, 3.60477021559837359669690449450, 5.66996440415567541345914817885, 7.18844300795727636685253122061, 8.335472794046371795066788145602, 9.611306223911925914743578756541, 10.41583186749910432522877753072, 11.24871844499723407833366241462, 12.49960865430350378168108841854, 13.54530382917637600444136471302

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