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Properties

Label 2-108-27.13-c1-0-0
Degree $2$
Conductor $108$
Sign $0.852 - 0.523i$
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  + (−1.73 + 0.0827i)3-s + (2.26 + 1.89i)5-s + (2.50 + 0.913i)7-s + (2.98 − 0.286i)9-s + (−2.22 + 1.86i)11-s + (0.588 − 3.33i)13-s + (−4.07 − 3.09i)15-s + (−2.40 + 4.16i)17-s + (−3.13 − 5.43i)19-s + (−4.41 − 1.37i)21-s + (0.841 − 0.306i)23-s + (0.648 + 3.67i)25-s + (−5.14 + 0.742i)27-s + (−1.06 − 6.01i)29-s + (7.86 − 2.86i)31-s + ⋯
L(s)  = 1  + (−0.998 + 0.0477i)3-s + (1.01 + 0.849i)5-s + (0.948 + 0.345i)7-s + (0.995 − 0.0954i)9-s + (−0.670 + 0.562i)11-s + (0.163 − 0.925i)13-s + (−1.05 − 0.800i)15-s + (−0.583 + 1.01i)17-s + (−0.719 − 1.24i)19-s + (−0.963 − 0.299i)21-s + (0.175 − 0.0638i)23-s + (0.129 + 0.735i)25-s + (−0.989 + 0.142i)27-s + (−0.196 − 1.11i)29-s + (1.41 − 0.513i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.852 - 0.523i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.852 - 0.523i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(108\)    =    \(2^{2} \cdot 3^{3}\)
Sign: $0.852 - 0.523i$
Analytic conductor: \(0.862384\)
Root analytic conductor: \(0.928646\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{108} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 108,\ (\ :1/2),\ 0.852 - 0.523i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.903275 + 0.255176i\)
\(L(\frac12)\) \(\approx\) \(0.903275 + 0.255176i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (1.73 - 0.0827i)T \)
good5 \( 1 + (-2.26 - 1.89i)T + (0.868 + 4.92i)T^{2} \)
7 \( 1 + (-2.50 - 0.913i)T + (5.36 + 4.49i)T^{2} \)
11 \( 1 + (2.22 - 1.86i)T + (1.91 - 10.8i)T^{2} \)
13 \( 1 + (-0.588 + 3.33i)T + (-12.2 - 4.44i)T^{2} \)
17 \( 1 + (2.40 - 4.16i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.13 + 5.43i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.841 + 0.306i)T + (17.6 - 14.7i)T^{2} \)
29 \( 1 + (1.06 + 6.01i)T + (-27.2 + 9.91i)T^{2} \)
31 \( 1 + (-7.86 + 2.86i)T + (23.7 - 19.9i)T^{2} \)
37 \( 1 + (4.71 - 8.15i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-1.77 + 10.0i)T + (-38.5 - 14.0i)T^{2} \)
43 \( 1 + (1.13 - 0.949i)T + (7.46 - 42.3i)T^{2} \)
47 \( 1 + (5.41 + 1.97i)T + (36.0 + 30.2i)T^{2} \)
53 \( 1 + 4.51T + 53T^{2} \)
59 \( 1 + (-8.89 - 7.45i)T + (10.2 + 58.1i)T^{2} \)
61 \( 1 + (0.667 + 0.243i)T + (46.7 + 39.2i)T^{2} \)
67 \( 1 + (-0.472 + 2.67i)T + (-62.9 - 22.9i)T^{2} \)
71 \( 1 + (-0.000646 + 0.00111i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-0.878 - 1.52i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-1.73 - 9.84i)T + (-74.2 + 27.0i)T^{2} \)
83 \( 1 + (0.296 + 1.68i)T + (-77.9 + 28.3i)T^{2} \)
89 \( 1 + (6.52 + 11.2i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (2.43 - 2.04i)T + (16.8 - 95.5i)T^{2} \)
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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

−13.62627419638423992971126946409, −12.81632650805357071739112777948, −11.49758345952041056876448555199, −10.62505092875279180806063698581, −9.988257355835266548901770523035, −8.283317328696890244025964294521, −6.82154515459085281154421292212, −5.82815142117319375832842363080, −4.70740270631443717530543802347, −2.24127936895244368630212664532, 1.58441627583088255445956406892, 4.58821269443696033360173741864, 5.42904748283713588172108781103, 6.65424257584834874631591524313, 8.188198748007217690335277181298, 9.443519614463884442945644635365, 10.60021532432554253489529706456, 11.46138199032523663011484600514, 12.59797458254539550328148180640, 13.51377925429530341343160564982

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