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Properties

Label 2-119-17.8-c1-0-0
Degree $2$
Conductor $119$
Sign $-0.997 - 0.0727i$
Analytic cond. $0.950219$
Root an. cond. $0.974792$
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  + (−0.686 + 0.686i)2-s + (−0.839 + 2.02i)3-s + 1.05i·4-s + (−2.58 − 1.07i)5-s + (−0.815 − 1.96i)6-s + (0.923 − 0.382i)7-s + (−2.09 − 2.09i)8-s + (−1.28 − 1.28i)9-s + (2.50 − 1.03i)10-s + (−0.577 − 1.39i)11-s + (−2.14 − 0.887i)12-s + 5.94i·13-s + (−0.371 + 0.897i)14-s + (4.33 − 4.33i)15-s + 0.769·16-s + (0.995 + 4.00i)17-s + ⋯
L(s)  = 1  + (−0.485 + 0.485i)2-s + (−0.484 + 1.17i)3-s + 0.528i·4-s + (−1.15 − 0.478i)5-s + (−0.332 − 0.803i)6-s + (0.349 − 0.144i)7-s + (−0.742 − 0.742i)8-s + (−0.427 − 0.427i)9-s + (0.793 − 0.328i)10-s + (−0.174 − 0.420i)11-s + (−0.618 − 0.256i)12-s + 1.64i·13-s + (−0.0993 + 0.239i)14-s + (1.12 − 1.12i)15-s + 0.192·16-s + (0.241 + 0.970i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 119 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 - 0.0727i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 119 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.997 - 0.0727i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(119\)    =    \(7 \cdot 17\)
Sign: $-0.997 - 0.0727i$
Analytic conductor: \(0.950219\)
Root analytic conductor: \(0.974792\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{119} (8, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 119,\ (\ :1/2),\ -0.997 - 0.0727i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0175600 + 0.481906i\)
\(L(\frac12)\) \(\approx\) \(0.0175600 + 0.481906i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-0.923 + 0.382i)T \)
17 \( 1 + (-0.995 - 4.00i)T \)
good2 \( 1 + (0.686 - 0.686i)T - 2iT^{2} \)
3 \( 1 + (0.839 - 2.02i)T + (-2.12 - 2.12i)T^{2} \)
5 \( 1 + (2.58 + 1.07i)T + (3.53 + 3.53i)T^{2} \)
11 \( 1 + (0.577 + 1.39i)T + (-7.77 + 7.77i)T^{2} \)
13 \( 1 - 5.94iT - 13T^{2} \)
19 \( 1 + (3.51 - 3.51i)T - 19iT^{2} \)
23 \( 1 + (-0.520 - 1.25i)T + (-16.2 + 16.2i)T^{2} \)
29 \( 1 + (-8.02 - 3.32i)T + (20.5 + 20.5i)T^{2} \)
31 \( 1 + (-3.27 + 7.90i)T + (-21.9 - 21.9i)T^{2} \)
37 \( 1 + (-0.502 + 1.21i)T + (-26.1 - 26.1i)T^{2} \)
41 \( 1 + (-0.619 + 0.256i)T + (28.9 - 28.9i)T^{2} \)
43 \( 1 + (-4.44 - 4.44i)T + 43iT^{2} \)
47 \( 1 - 2.88iT - 47T^{2} \)
53 \( 1 + (4.65 - 4.65i)T - 53iT^{2} \)
59 \( 1 + (7.19 + 7.19i)T + 59iT^{2} \)
61 \( 1 + (-4.53 + 1.87i)T + (43.1 - 43.1i)T^{2} \)
67 \( 1 - 8.44T + 67T^{2} \)
71 \( 1 + (1.81 - 4.39i)T + (-50.2 - 50.2i)T^{2} \)
73 \( 1 + (-9.28 - 3.84i)T + (51.6 + 51.6i)T^{2} \)
79 \( 1 + (1.67 + 4.04i)T + (-55.8 + 55.8i)T^{2} \)
83 \( 1 + (1.63 - 1.63i)T - 83iT^{2} \)
89 \( 1 - 2.29iT - 89T^{2} \)
97 \( 1 + (10.1 + 4.19i)T + (68.5 + 68.5i)T^{2} \)
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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

−14.31932512281350896541872704685, −12.68020021564325703979932553300, −11.76677713711589763017818336360, −10.95315767688581193390721448394, −9.647671180832921741149906557713, −8.557650843069904654575272223515, −7.80703236973440885859657900554, −6.30977552614211194971594052516, −4.47517971529466342257220614776, −3.87337481654624600381155893450, 0.65895760304589267125076538643, 2.71100876913990853648109853813, 5.04545104606198684338637212547, 6.49019178879242022372971942017, 7.55576698046980186516373068314, 8.495436486183096924949812439411, 10.16729226064607492231649349655, 11.04364802248768806610289227208, 11.89914128223276574815483464645, 12.58819291474586865529021349967

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