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Properties

Label 2-117-9.5-c2-0-0
Degree $2$
Conductor $117$
Sign $-0.391 + 0.920i$
Analytic cond. $3.18801$
Root an. cond. $1.78550$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.30 + 0.750i)2-s + (−0.0792 + 2.99i)3-s + (−0.873 + 1.51i)4-s + (−6.06 − 3.50i)5-s + (−2.14 − 3.95i)6-s + (0.489 + 0.847i)7-s − 8.62i·8-s + (−8.98 − 0.475i)9-s + 10.5·10-s + (12.1 − 7.00i)11-s + (−4.46 − 2.73i)12-s + (−1.80 + 3.12i)13-s + (−1.27 − 0.734i)14-s + (10.9 − 17.9i)15-s + (2.98 + 5.16i)16-s + 4.79i·17-s + ⋯
L(s)  = 1  + (−0.650 + 0.375i)2-s + (−0.0264 + 0.999i)3-s + (−0.218 + 0.378i)4-s + (−1.21 − 0.700i)5-s + (−0.357 − 0.659i)6-s + (0.0699 + 0.121i)7-s − 1.07i·8-s + (−0.998 − 0.0528i)9-s + 1.05·10-s + (1.10 − 0.637i)11-s + (−0.372 − 0.228i)12-s + (−0.138 + 0.240i)13-s + (−0.0909 − 0.0524i)14-s + (0.732 − 1.19i)15-s + (0.186 + 0.322i)16-s + 0.282i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.391 + 0.920i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.391 + 0.920i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(117\)    =    \(3^{2} \cdot 13\)
Sign: $-0.391 + 0.920i$
Analytic conductor: \(3.18801\)
Root analytic conductor: \(1.78550\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{117} (14, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 117,\ (\ :1),\ -0.391 + 0.920i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.00926876 - 0.0140111i\)
\(L(\frac12)\) \(\approx\) \(0.00926876 - 0.0140111i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.0792 - 2.99i)T \)
13 \( 1 + (1.80 - 3.12i)T \)
good2 \( 1 + (1.30 - 0.750i)T + (2 - 3.46i)T^{2} \)
5 \( 1 + (6.06 + 3.50i)T + (12.5 + 21.6i)T^{2} \)
7 \( 1 + (-0.489 - 0.847i)T + (-24.5 + 42.4i)T^{2} \)
11 \( 1 + (-12.1 + 7.00i)T + (60.5 - 104. i)T^{2} \)
17 \( 1 - 4.79iT - 289T^{2} \)
19 \( 1 + 29.6T + 361T^{2} \)
23 \( 1 + (38.6 + 22.3i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + (40.2 - 23.2i)T + (420.5 - 728. i)T^{2} \)
31 \( 1 + (1.40 - 2.43i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 - 9.83T + 1.36e3T^{2} \)
41 \( 1 + (11.8 + 6.85i)T + (840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (-38.8 - 67.2i)T + (-924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (19.6 - 11.3i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 - 35.0iT - 2.80e3T^{2} \)
59 \( 1 + (57.8 + 33.4i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (20.1 + 34.8i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-9.77 + 16.9i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 84.0iT - 5.04e3T^{2} \)
73 \( 1 + 6.25T + 5.32e3T^{2} \)
79 \( 1 + (-1.65 - 2.86i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (-73.3 + 42.3i)T + (3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 - 60.2iT - 7.92e3T^{2} \)
97 \( 1 + (9.35 + 16.2i)T + (-4.70e3 + 8.14e3i)T^{2} \)
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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

−14.31409713682285939523113109913, −12.70008400796109159959655549475, −11.88081260738779772339232106912, −10.81662788905081735905231952386, −9.407958258485156983807701159453, −8.631847722706881886820542279386, −7.995813698062716928610363685937, −6.31742888650884184846775252236, −4.37611914280329290387979176551, −3.79071844123557238769233111201, 0.01460482790123195742642921086, 1.99524284511032688292320699900, 4.02312055163835756605612756502, 6.01985434686661234776880117903, 7.29653622621147570933216040449, 8.120204946515953368099227601078, 9.320662560797103146487722483282, 10.66160639357385425528614936090, 11.55806562911771442427510790826, 12.20143824536747833742867533875

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