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Properties

Label 2-117-3.2-c2-0-3
Degree $2$
Conductor $117$
Sign $0.577 - 0.816i$
Analytic cond. $3.18801$
Root an. cond. $1.78550$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.65i·2-s + 1.26·4-s − 2.99i·5-s + 9.10·7-s + 8.70i·8-s + 4.94·10-s + 0.315i·11-s + 3.60·13-s + 15.0i·14-s − 9.31·16-s + 16.4i·17-s − 23.4·19-s − 3.79i·20-s − 0.520·22-s − 13.6i·23-s + ⋯
L(s)  = 1  + 0.826i·2-s + 0.317·4-s − 0.598i·5-s + 1.30·7-s + 1.08i·8-s + 0.494·10-s + 0.0286i·11-s + 0.277·13-s + 1.07i·14-s − 0.582·16-s + 0.968i·17-s − 1.23·19-s − 0.189i·20-s − 0.0236·22-s − 0.591i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.577 - 0.816i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(117\)    =    \(3^{2} \cdot 13\)
Sign: $0.577 - 0.816i$
Analytic conductor: \(3.18801\)
Root analytic conductor: \(1.78550\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{117} (53, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 117,\ (\ :1),\ 0.577 - 0.816i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.51228 + 0.782818i\)
\(L(\frac12)\) \(\approx\) \(1.51228 + 0.782818i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
13 \( 1 - 3.60T \)
good2 \( 1 - 1.65iT - 4T^{2} \)
5 \( 1 + 2.99iT - 25T^{2} \)
7 \( 1 - 9.10T + 49T^{2} \)
11 \( 1 - 0.315iT - 121T^{2} \)
17 \( 1 - 16.4iT - 289T^{2} \)
19 \( 1 + 23.4T + 361T^{2} \)
23 \( 1 + 13.6iT - 529T^{2} \)
29 \( 1 + 42.6iT - 841T^{2} \)
31 \( 1 + 4.47T + 961T^{2} \)
37 \( 1 + 23.2T + 1.36e3T^{2} \)
41 \( 1 + 24.6iT - 1.68e3T^{2} \)
43 \( 1 + 57.0T + 1.84e3T^{2} \)
47 \( 1 + 44.7iT - 2.20e3T^{2} \)
53 \( 1 + 61.8iT - 2.80e3T^{2} \)
59 \( 1 - 90.7iT - 3.48e3T^{2} \)
61 \( 1 + 89.6T + 3.72e3T^{2} \)
67 \( 1 - 98.0T + 4.48e3T^{2} \)
71 \( 1 - 77.6iT - 5.04e3T^{2} \)
73 \( 1 + 140.T + 5.32e3T^{2} \)
79 \( 1 - 71.5T + 6.24e3T^{2} \)
83 \( 1 - 109. iT - 6.88e3T^{2} \)
89 \( 1 - 119. iT - 7.92e3T^{2} \)
97 \( 1 - 39.1T + 9.40e3T^{2} \)
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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

−13.65592244977391340068967209040, −12.40062830294307073713864230053, −11.36312490891644650776311689660, −10.46806047538254212377112950140, −8.546426037653768069753511482434, −8.206119374759121010580624815967, −6.79498509918546222205583538686, −5.58940017713557377948592290467, −4.42886907609107245471618039841, −1.92622490768769510406182049736, 1.68324677615564069577048129029, 3.16642489158974538752024676309, 4.80294673842849022207922425114, 6.53596274300372422343518707257, 7.62817362922471084095810662915, 8.998272427369001772839623651203, 10.45346677892456457631871690375, 11.04990259556906846479882186340, 11.81511382919292982311510514362, 12.92285516112084383400204232541

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