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Properties

Label 2-117-117.32-c1-0-6
Degree $2$
Conductor $117$
Sign $0.884 + 0.467i$
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  + (0.286 + 0.286i)2-s + (−1.72 + 0.147i)3-s − 1.83i·4-s + (2.00 − 0.537i)5-s + (−0.536 − 0.452i)6-s + (1.90 − 0.510i)7-s + (1.09 − 1.09i)8-s + (2.95 − 0.508i)9-s + (0.728 + 0.420i)10-s + (−1.96 + 1.96i)11-s + (0.270 + 3.16i)12-s + (−0.908 − 3.48i)13-s + (0.692 + 0.399i)14-s + (−3.38 + 1.22i)15-s − 3.04·16-s + (2.88 + 5.00i)17-s + ⋯
L(s)  = 1  + (0.202 + 0.202i)2-s + (−0.996 + 0.0850i)3-s − 0.917i·4-s + (0.897 − 0.240i)5-s + (−0.219 − 0.184i)6-s + (0.720 − 0.192i)7-s + (0.388 − 0.388i)8-s + (0.985 − 0.169i)9-s + (0.230 + 0.133i)10-s + (−0.593 + 0.593i)11-s + (0.0781 + 0.914i)12-s + (−0.252 − 0.967i)13-s + (0.184 + 0.106i)14-s + (−0.873 + 0.315i)15-s − 0.760·16-s + (0.700 + 1.21i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.884 + 0.467i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.884 + 0.467i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(117\)    =    \(3^{2} \cdot 13\)
Sign: $0.884 + 0.467i$
Analytic conductor: \(0.934249\)
Root analytic conductor: \(0.966565\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{117} (32, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 117,\ (\ :1/2),\ 0.884 + 0.467i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.00026 - 0.248028i\)
\(L(\frac12)\) \(\approx\) \(1.00026 - 0.248028i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.72 - 0.147i)T \)
13 \( 1 + (0.908 + 3.48i)T \)
good2 \( 1 + (-0.286 - 0.286i)T + 2iT^{2} \)
5 \( 1 + (-2.00 + 0.537i)T + (4.33 - 2.5i)T^{2} \)
7 \( 1 + (-1.90 + 0.510i)T + (6.06 - 3.5i)T^{2} \)
11 \( 1 + (1.96 - 1.96i)T - 11iT^{2} \)
17 \( 1 + (-2.88 - 5.00i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (4.00 + 1.07i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (-1.46 - 2.53i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 0.0272iT - 29T^{2} \)
31 \( 1 + (-1.20 - 4.49i)T + (-26.8 + 15.5i)T^{2} \)
37 \( 1 + (-6.63 + 1.77i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (2.43 - 9.08i)T + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (-5.43 - 3.13i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (4.26 + 1.14i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + 8.65iT - 53T^{2} \)
59 \( 1 + (9.73 - 9.73i)T - 59iT^{2} \)
61 \( 1 + (-4.64 + 8.03i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-6.40 - 1.71i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + (-2.46 + 9.20i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 + (-0.696 - 0.696i)T + 73iT^{2} \)
79 \( 1 + (6.46 + 11.2i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-0.860 + 3.21i)T + (-71.8 - 41.5i)T^{2} \)
89 \( 1 + (0.569 + 2.12i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (-3.16 - 11.8i)T + (-84.0 + 48.5i)T^{2} \)
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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

−13.29086044676709964388061217632, −12.65347439271179153734837029342, −11.11484304678304181230454143715, −10.35771186354953683817906894888, −9.667159029869431383497758404646, −7.83168882011588188841258692292, −6.36161178117628431637606503911, −5.48024669524744598001209871371, −4.66396655716401626653109808096, −1.56241430122057494693486992964, 2.33019279804636593001361459552, 4.42616220718043957448603802669, 5.59882147710500159847094961101, 6.89082428178565014266073613492, 8.086850299622135905980812974104, 9.534751926806719181208559639906, 10.81101908048335376621951461121, 11.58504009892411023944748620382, 12.45968116128234964148502552360, 13.49497659918031376516689893942

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