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Properties

Label 2-117-117.49-c1-0-2
Degree $2$
Conductor $117$
Sign $0.540 - 0.841i$
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  + (1.67 + 0.967i)2-s + (−1.73 − 0.0524i)3-s + (0.871 + 1.50i)4-s + (2.26 + 1.30i)5-s + (−2.84 − 1.76i)6-s + 2.32i·7-s − 0.497i·8-s + (2.99 + 0.181i)9-s + (2.53 + 4.38i)10-s + (−4.66 − 2.69i)11-s + (−1.42 − 2.65i)12-s + (−1.31 − 3.35i)13-s + (−2.25 + 3.90i)14-s + (−3.85 − 2.38i)15-s + (2.22 − 3.85i)16-s + (0.835 − 1.44i)17-s + ⋯
L(s)  = 1  + (1.18 + 0.683i)2-s + (−0.999 − 0.0302i)3-s + (0.435 + 0.754i)4-s + (1.01 + 0.585i)5-s + (−1.16 − 0.719i)6-s + 0.880i·7-s − 0.175i·8-s + (0.998 + 0.0605i)9-s + (0.800 + 1.38i)10-s + (−1.40 − 0.812i)11-s + (−0.412 − 0.767i)12-s + (−0.366 − 0.930i)13-s + (−0.602 + 1.04i)14-s + (−0.995 − 0.615i)15-s + (0.556 − 0.963i)16-s + (0.202 − 0.351i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.540 - 0.841i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.540 - 0.841i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(117\)    =    \(3^{2} \cdot 13\)
Sign: $0.540 - 0.841i$
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: \(0\)
Selberg data: \((2,\ 117,\ (\ :1/2),\ 0.540 - 0.841i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.32653 + 0.724240i\)
\(L(\frac12)\) \(\approx\) \(1.32653 + 0.724240i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.73 + 0.0524i)T \)
13 \( 1 + (1.31 + 3.35i)T \)
good2 \( 1 + (-1.67 - 0.967i)T + (1 + 1.73i)T^{2} \)
5 \( 1 + (-2.26 - 1.30i)T + (2.5 + 4.33i)T^{2} \)
7 \( 1 - 2.32iT - 7T^{2} \)
11 \( 1 + (4.66 + 2.69i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (-0.835 + 1.44i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.90 - 1.09i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + 4.20T + 23T^{2} \)
29 \( 1 + (2.86 - 4.95i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-7.55 - 4.35i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (1.22 - 0.707i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 - 1.52iT - 41T^{2} \)
43 \( 1 - 1.87T + 43T^{2} \)
47 \( 1 + (4.47 - 2.58i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + 9.81T + 53T^{2} \)
59 \( 1 + (-6.59 + 3.80i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + 0.934T + 61T^{2} \)
67 \( 1 - 2.06iT - 67T^{2} \)
71 \( 1 + (10.9 + 6.33i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 - 12.6iT - 73T^{2} \)
79 \( 1 + (-3.46 - 6.00i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-7.95 + 4.59i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (-8.62 + 4.97i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 2.35iT - 97T^{2} \)
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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

−13.65111305469742957289923205089, −12.87723232804858255165669552295, −11.99896632143911474525815798617, −10.59690611472968018797929807808, −9.838742422422758899305743672476, −7.82770182546127502434865868231, −6.45373917833328885835314751871, −5.62889259985791955616401498338, −5.13303023783236174181216528882, −2.93383560022990523218683437339, 2.01543149161924208781164402229, 4.27608720021242016642176760662, 5.07800700022999542993751229556, 6.10915612479576197141167751801, 7.64686198680437349400644367036, 9.754567278707335321998339446735, 10.40669720665300417183274236502, 11.57010553904228217936695315096, 12.46508478279473700112932819792, 13.31822568946150212657307382224

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