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Properties

Label 2-108-12.11-c1-0-6
Degree $2$
Conductor $108$
Sign $0.866 + 0.5i$
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  + (1.22 − 0.707i)2-s + (0.999 − 1.73i)4-s + 2.82i·5-s − 1.73i·7-s − 2.82i·8-s + (2.00 + 3.46i)10-s − 4.89·11-s − 13-s + (−1.22 − 2.12i)14-s + (−2.00 − 3.46i)16-s + 2.82i·17-s + 5.19i·19-s + (4.89 + 2.82i)20-s + (−5.99 + 3.46i)22-s + 4.89·23-s + ⋯
L(s)  = 1  + (0.866 − 0.499i)2-s + (0.499 − 0.866i)4-s + 1.26i·5-s − 0.654i·7-s − 0.999i·8-s + (0.632 + 1.09i)10-s − 1.47·11-s − 0.277·13-s + (−0.327 − 0.566i)14-s + (−0.500 − 0.866i)16-s + 0.685i·17-s + 1.19i·19-s + (1.09 + 0.632i)20-s + (−1.27 + 0.738i)22-s + 1.02·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.866 + 0.5i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.866 + 0.5i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(108\)    =    \(2^{2} \cdot 3^{3}\)
Sign: $0.866 + 0.5i$
Analytic conductor: \(0.862384\)
Root analytic conductor: \(0.928646\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{108} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 108,\ (\ :1/2),\ 0.866 + 0.5i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.46275 - 0.391945i\)
\(L(\frac12)\) \(\approx\) \(1.46275 - 0.391945i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.22 + 0.707i)T \)
3 \( 1 \)
good5 \( 1 - 2.82iT - 5T^{2} \)
7 \( 1 + 1.73iT - 7T^{2} \)
11 \( 1 + 4.89T + 11T^{2} \)
13 \( 1 + T + 13T^{2} \)
17 \( 1 - 2.82iT - 17T^{2} \)
19 \( 1 - 5.19iT - 19T^{2} \)
23 \( 1 - 4.89T + 23T^{2} \)
29 \( 1 + 5.65iT - 29T^{2} \)
31 \( 1 + 3.46iT - 31T^{2} \)
37 \( 1 + T + 37T^{2} \)
41 \( 1 + 5.65iT - 41T^{2} \)
43 \( 1 - 3.46iT - 43T^{2} \)
47 \( 1 + 4.89T + 47T^{2} \)
53 \( 1 + 5.65iT - 53T^{2} \)
59 \( 1 - 4.89T + 59T^{2} \)
61 \( 1 - 11T + 61T^{2} \)
67 \( 1 - 12.1iT - 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + T + 73T^{2} \)
79 \( 1 + 1.73iT - 79T^{2} \)
83 \( 1 - 9.79T + 83T^{2} \)
89 \( 1 - 2.82iT - 89T^{2} \)
97 \( 1 + 13T + 97T^{2} \)
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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

−13.56591702545586773458051875234, −12.77112879581487018740784099000, −11.42675846969452617444052018382, −10.52750906299309664461428638242, −10.03972817666480461842307116961, −7.80511589406951707490348315572, −6.73714895808816940727636904501, −5.46966003214090417135767742386, −3.83954399237016636898470666470, −2.53358746501207533779415162693, 2.77828702923805117006288224857, 4.90535018477208437957651904602, 5.28584919165466854143316822829, 7.03252381209453066427768669185, 8.281721265933742721789710956170, 9.163159153657917161863809723513, 10.93906277710774321682413148161, 12.14263861958532953022595243991, 12.89531606599519667157622523628, 13.53462085649286161955794118302

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