Properties

Label 2-108-108.95-c1-0-15
Degree $2$
Conductor $108$
Sign $-0.723 + 0.690i$
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  + (0.0763 − 1.41i)2-s + (−0.463 − 1.66i)3-s + (−1.98 − 0.215i)4-s + (1.24 − 1.47i)5-s + (−2.39 + 0.526i)6-s + (−0.767 + 2.10i)7-s + (−0.456 + 2.79i)8-s + (−2.57 + 1.54i)9-s + (−1.99 − 1.86i)10-s + (2.76 − 2.32i)11-s + (0.560 + 3.41i)12-s + (0.799 − 4.53i)13-s + (2.91 + 1.24i)14-s + (−3.04 − 1.38i)15-s + (3.90 + 0.857i)16-s + (3.65 + 2.11i)17-s + ⋯
L(s)  = 1  + (0.0539 − 0.998i)2-s + (−0.267 − 0.963i)3-s + (−0.994 − 0.107i)4-s + (0.555 − 0.661i)5-s + (−0.976 + 0.214i)6-s + (−0.290 + 0.797i)7-s + (−0.161 + 0.986i)8-s + (−0.856 + 0.515i)9-s + (−0.630 − 0.590i)10-s + (0.833 − 0.699i)11-s + (0.161 + 0.986i)12-s + (0.221 − 1.25i)13-s + (0.780 + 0.332i)14-s + (−0.785 − 0.358i)15-s + (0.976 + 0.214i)16-s + (0.886 + 0.511i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.723 + 0.690i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.723 + 0.690i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(108\)    =    \(2^{2} \cdot 3^{3}\)
Sign: $-0.723 + 0.690i$
Analytic conductor: \(0.862384\)
Root analytic conductor: \(0.928646\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{108} (95, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 108,\ (\ :1/2),\ -0.723 + 0.690i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.349360 - 0.871860i\)
\(L(\frac12)\) \(\approx\) \(0.349360 - 0.871860i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.0763 + 1.41i)T \)
3 \( 1 + (0.463 + 1.66i)T \)
good5 \( 1 + (-1.24 + 1.47i)T + (-0.868 - 4.92i)T^{2} \)
7 \( 1 + (0.767 - 2.10i)T + (-5.36 - 4.49i)T^{2} \)
11 \( 1 + (-2.76 + 2.32i)T + (1.91 - 10.8i)T^{2} \)
13 \( 1 + (-0.799 + 4.53i)T + (-12.2 - 4.44i)T^{2} \)
17 \( 1 + (-3.65 - 2.11i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.84 - 2.21i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-4.90 + 1.78i)T + (17.6 - 14.7i)T^{2} \)
29 \( 1 + (-0.836 + 0.147i)T + (27.2 - 9.91i)T^{2} \)
31 \( 1 + (-0.155 - 0.428i)T + (-23.7 + 19.9i)T^{2} \)
37 \( 1 + (2.56 - 4.44i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (11.2 + 1.98i)T + (38.5 + 14.0i)T^{2} \)
43 \( 1 + (-6.14 - 7.32i)T + (-7.46 + 42.3i)T^{2} \)
47 \( 1 + (6.43 + 2.34i)T + (36.0 + 30.2i)T^{2} \)
53 \( 1 - 3.31iT - 53T^{2} \)
59 \( 1 + (-2.60 - 2.18i)T + (10.2 + 58.1i)T^{2} \)
61 \( 1 + (9.98 + 3.63i)T + (46.7 + 39.2i)T^{2} \)
67 \( 1 + (-0.436 - 0.0770i)T + (62.9 + 22.9i)T^{2} \)
71 \( 1 + (-0.668 + 1.15i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-7.86 - 13.6i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (10.2 - 1.80i)T + (74.2 - 27.0i)T^{2} \)
83 \( 1 + (0.245 + 1.39i)T + (-77.9 + 28.3i)T^{2} \)
89 \( 1 + (-4.09 + 2.36i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-3.17 + 2.66i)T + (16.8 - 95.5i)T^{2} \)
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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

−12.86177592342250397542666340640, −12.53115291102890350929637616149, −11.44696218988263499135449558102, −10.29358074765804364422566570268, −8.923653367066203624481274978256, −8.244505548403694014772653406578, −6.14151094898036920597346918606, −5.29977803131769179519837889848, −3.10134093614318432130754945917, −1.34175891204674602804640455296, 3.70687391200902337311787662651, 4.84434737913253667916610616708, 6.37935605306337815797849767929, 7.06736412944498040682485566723, 8.930174915844502918678657596203, 9.708593397347945388555233249591, 10.62494321932428356379179017751, 12.00961676696787980797794974410, 13.56789319179077279049776479749, 14.34341500198215503883697160470

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