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Properties

Label 2-117-117.4-c1-0-7
Degree $2$
Conductor $117$
Sign $0.219 + 0.975i$
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  − 0.968i·2-s + (−0.767 − 1.55i)3-s + 1.06·4-s + (3.54 + 2.04i)5-s + (−1.50 + 0.743i)6-s + (−3.06 − 1.77i)7-s − 2.96i·8-s + (−1.82 + 2.38i)9-s + (1.98 − 3.43i)10-s + 2.27i·11-s + (−0.816 − 1.64i)12-s + (−3.37 − 1.27i)13-s + (−1.71 + 2.96i)14-s + (0.455 − 7.08i)15-s − 0.745·16-s + (1.48 + 2.56i)17-s + ⋯
L(s)  = 1  − 0.684i·2-s + (−0.443 − 0.896i)3-s + 0.531·4-s + (1.58 + 0.915i)5-s + (−0.613 + 0.303i)6-s + (−1.15 − 0.669i)7-s − 1.04i·8-s + (−0.606 + 0.794i)9-s + (0.627 − 1.08i)10-s + 0.685i·11-s + (−0.235 − 0.476i)12-s + (−0.935 − 0.353i)13-s + (−0.458 + 0.793i)14-s + (0.117 − 1.82i)15-s − 0.186·16-s + (0.359 + 0.622i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.219 + 0.975i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.219 + 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(117\)    =    \(3^{2} \cdot 13\)
Sign: $0.219 + 0.975i$
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: \((2,\ 117,\ (\ :1/2),\ 0.219 + 0.975i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.904920 - 0.724075i\)
\(L(\frac12)\) \(\approx\) \(0.904920 - 0.724075i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.767 + 1.55i)T \)
13 \( 1 + (3.37 + 1.27i)T \)
good2 \( 1 + 0.968iT - 2T^{2} \)
5 \( 1 + (-3.54 - 2.04i)T + (2.5 + 4.33i)T^{2} \)
7 \( 1 + (3.06 + 1.77i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 - 2.27iT - 11T^{2} \)
17 \( 1 + (-1.48 - 2.56i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.575 - 0.332i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.27 - 2.21i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 1.93T + 29T^{2} \)
31 \( 1 + (0.733 + 0.423i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (1.34 + 0.775i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (1.84 - 1.06i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-1.22 + 2.11i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (4.45 - 2.57i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 - 3.82T + 53T^{2} \)
59 \( 1 + 0.343iT - 59T^{2} \)
61 \( 1 + (-5.85 + 10.1i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (9.78 - 5.64i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-7.40 + 4.27i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 - 0.0644iT - 73T^{2} \)
79 \( 1 + (-1.74 - 3.01i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (5.65 - 3.26i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (13.1 + 7.58i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (14.4 + 8.33i)T + (48.5 + 84.0i)T^{2} \)
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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

−12.99611209635633945885859471899, −12.54000101570520367350573663174, −11.12204813856874176136860401112, −10.17928364340128747036678565599, −9.751864954960388382691284736372, −7.28316518837804849581365965794, −6.70712769272082357716905712335, −5.73848913167622276587317476626, −3.05057029424192748949958244330, −1.88434102266156859927859721486, 2.68381177902630943088114223809, 5.10647901196170373664772456789, 5.80945957102353344002993332011, 6.65637510970265397306075297760, 8.742049007844841703080952188237, 9.476029727865669137584939872741, 10.32257710754877876432915783502, 11.76526690206541981381498297980, 12.71950984398544115031403517807, 13.93795299966639876713757949355

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