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Properties

Label 2-117-117.49-c1-0-11
Degree $2$
Conductor $117$
Sign $-0.428 - 0.903i$
Analytic cond. $0.934249$
Root an. cond. $0.966565$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.5 − 0.866i)2-s + (−1.5 − 0.866i)3-s + (0.5 + 0.866i)4-s + (−1.5 − 0.866i)5-s + (1.5 + 2.59i)6-s + 1.73i·7-s + 1.73i·8-s + (1.5 + 2.59i)9-s + (1.5 + 2.59i)10-s + (−3 − 1.73i)11-s − 1.73i·12-s + (−2.5 + 2.59i)13-s + (1.49 − 2.59i)14-s + (1.5 + 2.59i)15-s + (2.49 − 4.33i)16-s + (−1.5 + 2.59i)17-s + ⋯
L(s)  = 1  + (−1.06 − 0.612i)2-s + (−0.866 − 0.499i)3-s + (0.250 + 0.433i)4-s + (−0.670 − 0.387i)5-s + (0.612 + 1.06i)6-s + 0.654i·7-s + 0.612i·8-s + (0.5 + 0.866i)9-s + (0.474 + 0.821i)10-s + (−0.904 − 0.522i)11-s − 0.500i·12-s + (−0.693 + 0.720i)13-s + (0.400 − 0.694i)14-s + (0.387 + 0.670i)15-s + (0.624 − 1.08i)16-s + (−0.363 + 0.630i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.428 - 0.903i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.428 - 0.903i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(117\)    =    \(3^{2} \cdot 13\)
Sign: $-0.428 - 0.903i$
Analytic conductor: \(0.934249\)
Root analytic conductor: \(0.966565\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{117} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 117,\ (\ :1/2),\ -0.428 - 0.903i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.5 + 0.866i)T \)
13 \( 1 + (2.5 - 2.59i)T \)
good2 \( 1 + (1.5 + 0.866i)T + (1 + 1.73i)T^{2} \)
5 \( 1 + (1.5 + 0.866i)T + (2.5 + 4.33i)T^{2} \)
7 \( 1 - 1.73iT - 7T^{2} \)
11 \( 1 + (3 + 1.73i)T + (5.5 + 9.52i)T^{2} \)
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 \( 1 + 3T + 23T^{2} \)
29 \( 1 + (-3 + 5.19i)T + (-14.5 - 25.1i)T^{2} \)
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 \( 1 - 12.1iT - 41T^{2} \)
43 \( 1 + T + 43T^{2} \)
47 \( 1 + (4.5 - 2.59i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 - 6T + 53T^{2} \)
59 \( 1 + (-3 + 1.73i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + 5T + 61T^{2} \)
67 \( 1 + 12.1iT - 67T^{2} \)
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 \( 1 - 15.5iT - 97T^{2} \)
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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

−12.35165226491345955772984980588, −11.66807984831276045420323313624, −10.84897546948938748494242978061, −9.780559707535456130152025476849, −8.480946020083334206735078351896, −7.72713435356032831754083300337, −6.05908241654551215166689785058, −4.75082974210359934621131928119, −2.15276262389819621973419206531, 0, 3.79158084802677213002512374174, 5.29878391325248697301060944214, 7.03439552964069623066767208973, 7.47458363107256614223360335633, 8.923219890284389932905477106741, 10.27377683308262605034854737643, 10.56959718220958168531791237660, 12.03435267932437668437033047366, 12.98354886104095708706834147321

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