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Properties

Label 2-117-117.20-c1-0-0
Degree $2$
Conductor $117$
Sign $-0.0602 - 0.998i$
Analytic cond. $0.934249$
Root an. cond. $0.966565$
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  + (−1.02 − 0.275i)2-s + (−1.73 + 0.00536i)3-s + (−0.747 − 0.431i)4-s + (2.22 + 0.595i)5-s + (1.78 + 0.472i)6-s + (−3.66 + 3.66i)7-s + (2.15 + 2.15i)8-s + (2.99 − 0.0185i)9-s + (−2.12 − 1.22i)10-s + (−0.708 + 2.64i)11-s + (1.29 + 0.743i)12-s + (−2.69 + 2.39i)13-s + (4.78 − 2.76i)14-s + (−3.85 − 1.01i)15-s + (−0.764 − 1.32i)16-s + (−0.0716 − 0.124i)17-s + ⋯
L(s)  = 1  + (−0.728 − 0.195i)2-s + (−0.999 + 0.00309i)3-s + (−0.373 − 0.215i)4-s + (0.993 + 0.266i)5-s + (0.728 + 0.192i)6-s + (−1.38 + 1.38i)7-s + (0.763 + 0.763i)8-s + (0.999 − 0.00619i)9-s + (−0.671 − 0.387i)10-s + (−0.213 + 0.797i)11-s + (0.374 + 0.214i)12-s + (−0.747 + 0.664i)13-s + (1.27 − 0.737i)14-s + (−0.994 − 0.263i)15-s + (−0.191 − 0.331i)16-s + (−0.0173 − 0.0300i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0602 - 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0602 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(117\)    =    \(3^{2} \cdot 13\)
Sign: $-0.0602 - 0.998i$
Analytic conductor: \(0.934249\)
Root analytic conductor: \(0.966565\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{117} (20, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 117,\ (\ :1/2),\ -0.0602 - 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.255758 + 0.271668i\)
\(L(\frac12)\) \(\approx\) \(0.255758 + 0.271668i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.73 - 0.00536i)T \)
13 \( 1 + (2.69 - 2.39i)T \)
good2 \( 1 + (1.02 + 0.275i)T + (1.73 + i)T^{2} \)
5 \( 1 + (-2.22 - 0.595i)T + (4.33 + 2.5i)T^{2} \)
7 \( 1 + (3.66 - 3.66i)T - 7iT^{2} \)
11 \( 1 + (0.708 - 2.64i)T + (-9.52 - 5.5i)T^{2} \)
17 \( 1 + (0.0716 + 0.124i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.552 + 2.06i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 - 0.843T + 23T^{2} \)
29 \( 1 + (-0.523 + 0.302i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (0.905 - 3.38i)T + (-26.8 - 15.5i)T^{2} \)
37 \( 1 + (-0.973 - 3.63i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + (5.53 - 5.53i)T - 41iT^{2} \)
43 \( 1 + 0.216iT - 43T^{2} \)
47 \( 1 + (-10.1 + 2.72i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 - 8.00iT - 53T^{2} \)
59 \( 1 + (-7.19 + 1.92i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + 1.35T + 61T^{2} \)
67 \( 1 + (-5.41 - 5.41i)T + 67iT^{2} \)
71 \( 1 + (-7.04 - 1.88i)T + (61.4 + 35.5i)T^{2} \)
73 \( 1 + (3.76 - 3.76i)T - 73iT^{2} \)
79 \( 1 + (-1.62 + 2.81i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (3.94 + 14.7i)T + (-71.8 + 41.5i)T^{2} \)
89 \( 1 + (-14.2 + 3.81i)T + (77.0 - 44.5i)T^{2} \)
97 \( 1 + (0.384 + 0.384i)T + 97iT^{2} \)
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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

−13.55179374779226921958271347477, −12.64803616129900607347726563857, −11.71457804718099417373654284329, −10.22544134324124806882411040354, −9.756852369191603898379801518154, −8.992886105577169873852770027172, −6.99027441796383237487896515636, −5.93906176054755214611274240389, −4.93954795101334370904259628374, −2.23753387386426308761858275196, 0.57955395697280556600847055186, 3.83038790592677598009065727992, 5.46243675510767464930568099660, 6.66741994761423500589561260182, 7.67119790295221443841111032024, 9.376464946810177573490934186101, 10.05252772650841819394016288526, 10.68136639686878911858834803366, 12.51381850182485642913915219396, 13.20608429269621993405266921701

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