Properties

Label 2-108-12.11-c1-0-3
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

\(L(1)\) \(\approx\) \(0.702105 - 0.188128i\)
\(L(\frac12)\) \(\approx\) \(0.702105 - 0.188128i\)
\(L(\frac{3}{2})\) 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

−13.88899356930377682860245414798, −12.41742676422328489781301743072, −11.51030860027074266517156018103, −10.07056945536151663804908059052, −9.231739868664361363571273147442, −8.285813732700835825090870302135, −7.10773983113878985981094762058, −5.77710144178911363071619859598, −4.31239716399917921215721847652, −1.28738494146991810369837546020, 2.31677886908070896022890697102, 3.78108429902194902533382754795, 6.29241448203311121119305129276, 7.16132092439863928675358722446, 8.581188624764473361244357411680, 9.579514382579032216601020781947, 10.63309385102304075552945923961, 11.56634646519630827357679526066, 12.31226909877379339742711803629, 13.84615297786431326546519009863

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