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Properties

Label 2-117-117.2-c1-0-8
Degree $2$
Conductor $117$
Sign $0.993 + 0.115i$
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  + (0.0604 + 0.0604i)2-s + (1.25 + 1.19i)3-s − 1.99i·4-s + (0.466 − 1.73i)5-s + (0.00379 + 0.148i)6-s + (0.132 − 0.495i)7-s + (0.241 − 0.241i)8-s + (0.153 + 2.99i)9-s + (0.133 − 0.0770i)10-s + (−4.05 + 4.05i)11-s + (2.37 − 2.50i)12-s + (3.18 + 1.69i)13-s + (0.0379 − 0.0219i)14-s + (2.66 − 1.62i)15-s − 3.95·16-s + (−2.73 + 4.74i)17-s + ⋯
L(s)  = 1  + (0.0427 + 0.0427i)2-s + (0.724 + 0.688i)3-s − 0.996i·4-s + (0.208 − 0.777i)5-s + (0.00154 + 0.0604i)6-s + (0.0501 − 0.187i)7-s + (0.0853 − 0.0853i)8-s + (0.0512 + 0.998i)9-s + (0.0421 − 0.0243i)10-s + (−1.22 + 1.22i)11-s + (0.686 − 0.722i)12-s + (0.882 + 0.470i)13-s + (0.0101 − 0.00585i)14-s + (0.686 − 0.420i)15-s − 0.989·16-s + (−0.663 + 1.14i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.993 + 0.115i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.993 + 0.115i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(117\)    =    \(3^{2} \cdot 13\)
Sign: $0.993 + 0.115i$
Analytic conductor: \(0.934249\)
Root analytic conductor: \(0.966565\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{117} (2, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 117,\ (\ :1/2),\ 0.993 + 0.115i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.28960 - 0.0747368i\)
\(L(\frac12)\) \(\approx\) \(1.28960 - 0.0747368i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.25 - 1.19i)T \)
13 \( 1 + (-3.18 - 1.69i)T \)
good2 \( 1 + (-0.0604 - 0.0604i)T + 2iT^{2} \)
5 \( 1 + (-0.466 + 1.73i)T + (-4.33 - 2.5i)T^{2} \)
7 \( 1 + (-0.132 + 0.495i)T + (-6.06 - 3.5i)T^{2} \)
11 \( 1 + (4.05 - 4.05i)T - 11iT^{2} \)
17 \( 1 + (2.73 - 4.74i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.92 + 7.17i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (-2.21 + 3.84i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 0.515iT - 29T^{2} \)
31 \( 1 + (3.18 + 0.854i)T + (26.8 + 15.5i)T^{2} \)
37 \( 1 + (0.815 - 3.04i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + (2.70 - 0.724i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (-1.81 + 1.04i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (1.08 + 4.06i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + 8.79iT - 53T^{2} \)
59 \( 1 + (-5.47 + 5.47i)T - 59iT^{2} \)
61 \( 1 + (-2.89 - 5.02i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.38 + 5.17i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (-2.49 + 0.668i)T + (61.4 - 35.5i)T^{2} \)
73 \( 1 + (4.21 + 4.21i)T + 73iT^{2} \)
79 \( 1 + (2.43 - 4.20i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (3.28 - 0.879i)T + (71.8 - 41.5i)T^{2} \)
89 \( 1 + (-10.8 - 2.90i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (-2.14 - 0.574i)T + (84.0 + 48.5i)T^{2} \)
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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

−13.39120812628868957771335246093, −13.00605123175846028858253890613, −10.98866557914320609776026790026, −10.36020197090063147477401554081, −9.236780196040483611209310789985, −8.498921561274459105588187683989, −6.84980320214240669311619255810, −5.16338534485585707097521360686, −4.41821767479356707826956675585, −2.11080751001140622447629088626, 2.62919217665138692461628391823, 3.52803974338421256512944670451, 5.87108081812158696272809983341, 7.20037719372858229362852700951, 8.107332619060199467474166594799, 8.916871559548734712861606458126, 10.57813897171626993627166757871, 11.57786925217473549341072003176, 12.80209404108918199020333320671, 13.48439079997079097828885918029

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