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Properties

Label 2-117-117.4-c1-0-4
Degree $2$
Conductor $117$
Sign $0.815 + 0.578i$
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  − 1.73i·2-s + 1.73i·3-s − 0.999·4-s + (1.5 + 0.866i)5-s + 2.99·6-s + (1.5 + 0.866i)7-s − 1.73i·8-s − 2.99·9-s + (1.49 − 2.59i)10-s − 3.46i·11-s − 1.73i·12-s + (−1 + 3.46i)13-s + (1.49 − 2.59i)14-s + (−1.49 + 2.59i)15-s − 5·16-s + (−1.5 − 2.59i)17-s + ⋯
L(s)  = 1  − 1.22i·2-s + 0.999i·3-s − 0.499·4-s + (0.670 + 0.387i)5-s + 1.22·6-s + (0.566 + 0.327i)7-s − 0.612i·8-s − 0.999·9-s + (0.474 − 0.821i)10-s − 1.04i·11-s − 0.499i·12-s + (−0.277 + 0.960i)13-s + (0.400 − 0.694i)14-s + (−0.387 + 0.670i)15-s − 1.25·16-s + (−0.363 − 0.630i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.815 + 0.578i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.815 + 0.578i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(117\)    =    \(3^{2} \cdot 13\)
Sign: $0.815 + 0.578i$
Analytic conductor: \(0.934249\)
Root analytic conductor: \(0.966565\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{117} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 117,\ (\ :1/2),\ 0.815 + 0.578i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.12652 - 0.359167i\)
\(L(\frac12)\) \(\approx\) \(1.12652 - 0.359167i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - 1.73iT \)
13 \( 1 + (1 - 3.46i)T \)
good2 \( 1 + 1.73iT - 2T^{2} \)
5 \( 1 + (-1.5 - 0.866i)T + (2.5 + 4.33i)T^{2} \)
7 \( 1 + (-1.5 - 0.866i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + 3.46iT - 11T^{2} \)
17 \( 1 + (1.5 + 2.59i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.5 - 0.866i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.5 - 2.59i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 6T + 29T^{2} \)
31 \( 1 + (-7.5 - 4.33i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (4.5 + 2.59i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (10.5 - 6.06i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.5 + 0.866i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-4.5 + 2.59i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 - 6T + 53T^{2} \)
59 \( 1 + 3.46iT - 59T^{2} \)
61 \( 1 + (-2.5 + 4.33i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-10.5 + 6.06i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (7.5 - 4.33i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 - 6.92iT - 73T^{2} \)
79 \( 1 + (-5.5 - 9.52i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-4.5 + 2.59i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (-13.5 - 7.79i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-13.5 - 7.79i)T + (48.5 + 84.0i)T^{2} \)
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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

−13.51607083454318894937519297152, −11.87309586014801198716934559275, −11.32979864950134251443571025590, −10.42249421526834460621705263113, −9.555554372129609893289075813831, −8.608430982151890089595969902945, −6.54227041416560991394916286023, −5.07752838344390214932622301790, −3.61107539215575108044696718396, −2.27075375258071963491569305690, 2.05794273235526551664889387787, 4.93825816823722969879644120524, 5.97787034747356742681129252949, 7.07519335918192037213891224748, 7.902841901973221556917586273794, 8.880287183576737153781596149957, 10.45909693767849444608675620981, 11.80430470715270512324443966518, 12.96275518142648054401127533643, 13.66759014459622925712490442218

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