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Properties

Label 2-117-117.49-c1-0-7
Degree $2$
Conductor $117$
Sign $0.916 + 0.399i$
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  + (0.677 + 0.391i)2-s + (1.22 − 1.22i)3-s + (−0.693 − 1.20i)4-s + (−0.0536 − 0.0309i)5-s + (1.30 − 0.350i)6-s + 3.75i·7-s − 2.65i·8-s + (0.00294 − 2.99i)9-s + (−0.0242 − 0.0419i)10-s + (−1.11 − 0.641i)11-s + (−2.32 − 0.623i)12-s + (1.65 + 3.20i)13-s + (−1.46 + 2.54i)14-s + (−0.103 + 0.0277i)15-s + (−0.349 + 0.606i)16-s + (−2.74 + 4.75i)17-s + ⋯
L(s)  = 1  + (0.479 + 0.276i)2-s + (0.707 − 0.706i)3-s + (−0.346 − 0.600i)4-s + (−0.0239 − 0.0138i)5-s + (0.534 − 0.142i)6-s + 1.41i·7-s − 0.937i·8-s + (0.000981 − 0.999i)9-s + (−0.00766 − 0.0132i)10-s + (−0.334 − 0.193i)11-s + (−0.669 − 0.179i)12-s + (0.458 + 0.888i)13-s + (−0.392 + 0.680i)14-s + (−0.0267 + 0.00715i)15-s + (−0.0874 + 0.151i)16-s + (−0.665 + 1.15i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.916 + 0.399i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.916 + 0.399i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(117\)    =    \(3^{2} \cdot 13\)
Sign: $0.916 + 0.399i$
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: \((2,\ 117,\ (\ :1/2),\ 0.916 + 0.399i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.40614 - 0.292794i\)
\(L(\frac12)\) \(\approx\) \(1.40614 - 0.292794i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.22 + 1.22i)T \)
13 \( 1 + (-1.65 - 3.20i)T \)
good2 \( 1 + (-0.677 - 0.391i)T + (1 + 1.73i)T^{2} \)
5 \( 1 + (0.0536 + 0.0309i)T + (2.5 + 4.33i)T^{2} \)
7 \( 1 - 3.75iT - 7T^{2} \)
11 \( 1 + (1.11 + 0.641i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (2.74 - 4.75i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.72 - 1.57i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + 5.65T + 23T^{2} \)
29 \( 1 + (-3.56 + 6.17i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-3.62 - 2.09i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (4.03 - 2.33i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + 9.88iT - 41T^{2} \)
43 \( 1 - 4.22T + 43T^{2} \)
47 \( 1 + (-2.73 + 1.57i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + 0.752T + 53T^{2} \)
59 \( 1 + (0.310 - 0.179i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 - 1.65T + 61T^{2} \)
67 \( 1 + 1.65iT - 67T^{2} \)
71 \( 1 + (-11.3 - 6.57i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + 6.09iT - 73T^{2} \)
79 \( 1 + (0.616 + 1.06i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (11.0 - 6.38i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (-1.30 + 0.751i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 10.0iT - 97T^{2} \)
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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

−13.76256716555292946547145446167, −12.57471298081809798053809552470, −11.83280897724671535335158581062, −10.10997595488099719844103809693, −8.981344523695094186965834730039, −8.221208340985478124957396971736, −6.48452773130196013261083492789, −5.78059996051276534789206589641, −4.02842527869058099065908088283, −2.10417009161551699469891480763, 2.97329917938861271450110741147, 4.04788822317999249439152781983, 5.07101708514880058375326143317, 7.34962396929017227501523923724, 8.157415612407302059202323760414, 9.439945350677639203368030876607, 10.50215624795836805070370004584, 11.44273381087145321233977084524, 12.94367458762988532117948298021, 13.69398841308266287006483803581

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