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Properties

Label 2-117-39.20-c1-0-3
Degree $2$
Conductor $117$
Sign $-0.608 + 0.793i$
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  + (−0.517 − 1.93i)2-s + (−1.73 + 0.999i)4-s + (1.93 − 1.93i)5-s + (−0.5 − 0.133i)7-s + (−4.73 − 2.73i)10-s + (−4.05 + 1.08i)11-s + (3.59 + 0.232i)13-s + 1.03i·14-s + (−1.99 + 3.46i)16-s + (3.34 + 5.79i)17-s + (1.63 − 6.09i)19-s + (−1.41 + 5.27i)20-s + (4.19 + 7.26i)22-s + (1.22 − 2.12i)23-s − 2.46i·25-s + (−1.41 − 7.07i)26-s + ⋯
L(s)  = 1  + (−0.366 − 1.36i)2-s + (−0.866 + 0.499i)4-s + (0.863 − 0.863i)5-s + (−0.188 − 0.0506i)7-s + (−1.49 − 0.863i)10-s + (−1.22 + 0.327i)11-s + (0.997 + 0.0643i)13-s + 0.276i·14-s + (−0.499 + 0.866i)16-s + (0.811 + 1.40i)17-s + (0.374 − 1.39i)19-s + (−0.316 + 1.18i)20-s + (0.894 + 1.54i)22-s + (0.255 − 0.442i)23-s − 0.492i·25-s + (−0.277 − 1.38i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.608 + 0.793i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.608 + 0.793i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(117\)    =    \(3^{2} \cdot 13\)
Sign: $-0.608 + 0.793i$
Analytic conductor: \(0.934249\)
Root analytic conductor: \(0.966565\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{117} (98, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 117,\ (\ :1/2),\ -0.608 + 0.793i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.415767 - 0.842663i\)
\(L(\frac12)\) \(\approx\) \(0.415767 - 0.842663i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
13 \( 1 + (-3.59 - 0.232i)T \)
good2 \( 1 + (0.517 + 1.93i)T + (-1.73 + i)T^{2} \)
5 \( 1 + (-1.93 + 1.93i)T - 5iT^{2} \)
7 \( 1 + (0.5 + 0.133i)T + (6.06 + 3.5i)T^{2} \)
11 \( 1 + (4.05 - 1.08i)T + (9.52 - 5.5i)T^{2} \)
17 \( 1 + (-3.34 - 5.79i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.63 + 6.09i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (-1.22 + 2.12i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.22 - 0.707i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-4.63 - 4.63i)T + 31iT^{2} \)
37 \( 1 + (-0.830 - 3.09i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + (-0.378 - 1.41i)T + (-35.5 + 20.5i)T^{2} \)
43 \( 1 + (6.69 - 3.86i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-2.31 - 2.31i)T + 47iT^{2} \)
53 \( 1 - 5.93iT - 53T^{2} \)
59 \( 1 + (0.0507 - 0.189i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (6.59 + 11.4i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (7.59 - 2.03i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (15.1 + 4.05i)T + (61.4 + 35.5i)T^{2} \)
73 \( 1 + (-2.90 + 2.90i)T - 73iT^{2} \)
79 \( 1 - 7.19T + 79T^{2} \)
83 \( 1 + (-5.27 + 5.27i)T - 83iT^{2} \)
89 \( 1 + (4.38 - 1.17i)T + (77.0 - 44.5i)T^{2} \)
97 \( 1 + (1.30 - 4.86i)T + (-84.0 - 48.5i)T^{2} \)
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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

−13.05425720487770923074810810988, −12.19185510385262916666200644502, −10.87337663195813735640050006781, −10.18111899116183032634693294121, −9.203391651495216453712060120488, −8.257226002161242438294590477859, −6.26495207537446421593941058629, −4.83591734425286094627685709194, −3.03532220408995346974800512811, −1.42332209345856480948375067618, 2.93243566888240504604187183158, 5.44543275254098123184577273261, 6.11435758023908675058791089821, 7.31754825431984126542739787284, 8.220526401950965889927726982274, 9.572589239706844068433331381714, 10.42723949047675937770578827198, 11.77024661832632330343526543458, 13.46744745710886438746686165618, 14.00790407366033647534538584358

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