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Properties

Label 2-117-117.25-c1-0-6
Degree $2$
Conductor $117$
Sign $0.512 - 0.858i$
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  + (1.41 + 0.818i)2-s + (0.471 + 1.66i)3-s + (0.338 + 0.587i)4-s + (0.950 − 0.548i)5-s + (−0.696 + 2.74i)6-s + (−2.77 − 1.60i)7-s − 2.16i·8-s + (−2.55 + 1.57i)9-s + 1.79·10-s + (−1.52 − 0.879i)11-s + (−0.818 + 0.841i)12-s + (2.19 + 2.85i)13-s + (−2.62 − 4.54i)14-s + (1.36 + 1.32i)15-s + (2.44 − 4.24i)16-s + 1.47·17-s + ⋯
L(s)  = 1  + (1.00 + 0.578i)2-s + (0.271 + 0.962i)3-s + (0.169 + 0.293i)4-s + (0.425 − 0.245i)5-s + (−0.284 + 1.12i)6-s + (−1.05 − 0.606i)7-s − 0.764i·8-s + (−0.852 + 0.523i)9-s + 0.567·10-s + (−0.459 − 0.265i)11-s + (−0.236 + 0.242i)12-s + (0.609 + 0.792i)13-s + (−0.701 − 1.21i)14-s + (0.351 + 0.342i)15-s + (0.612 − 1.06i)16-s + 0.357·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.512 - 0.858i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.512 - 0.858i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(117\)    =    \(3^{2} \cdot 13\)
Sign: $0.512 - 0.858i$
Analytic conductor: \(0.934249\)
Root analytic conductor: \(0.966565\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{117} (25, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 117,\ (\ :1/2),\ 0.512 - 0.858i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.44191 + 0.818211i\)
\(L(\frac12)\) \(\approx\) \(1.44191 + 0.818211i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.471 - 1.66i)T \)
13 \( 1 + (-2.19 - 2.85i)T \)
good2 \( 1 + (-1.41 - 0.818i)T + (1 + 1.73i)T^{2} \)
5 \( 1 + (-0.950 + 0.548i)T + (2.5 - 4.33i)T^{2} \)
7 \( 1 + (2.77 + 1.60i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (1.52 + 0.879i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 - 1.47T + 17T^{2} \)
19 \( 1 - 3.61iT - 19T^{2} \)
23 \( 1 + (2.34 + 4.06i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.959 + 1.66i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-5.68 + 3.28i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 - 11.6iT - 37T^{2} \)
41 \( 1 + (4.68 - 2.70i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.889 + 1.54i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (8.90 + 5.13i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 - 11.7T + 53T^{2} \)
59 \( 1 + (-4.78 + 2.76i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (0.985 - 1.70i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (7.15 - 4.13i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 5.84iT - 71T^{2} \)
73 \( 1 - 1.24iT - 73T^{2} \)
79 \( 1 + (0.242 - 0.420i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-13.4 - 7.75i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + 13.4iT - 89T^{2} \)
97 \( 1 + (5.15 + 2.97i)T + (48.5 + 84.0i)T^{2} \)
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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

−13.59101934575037704467514158984, −13.34954594555060816763278916138, −11.82137732025988187047388394727, −10.17102072454226074416422660558, −9.801880919461900046004828470731, −8.343066890789111454377178815418, −6.61662519539943514218391920635, −5.68473399868536440036075989183, −4.39556664768972248229406280059, −3.37160375601497057484405085312, 2.41753269211083985403222222254, 3.38740086623251928302607528661, 5.45513276967963559924379859223, 6.36762884441206797341728901517, 7.86497761064425204177669974713, 9.067766619255447455491980306363, 10.47035652818198299208248358800, 11.80041516338167663482063505104, 12.56388775918132419095644453540, 13.27092613955277725845116908133

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