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Properties

Label 2-119-17.8-c1-0-2
Degree $2$
Conductor $119$
Sign $0.413 - 0.910i$
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  + (0.315 − 0.315i)2-s + (−0.558 + 1.34i)3-s + 1.80i·4-s + (−0.410 − 0.169i)5-s + (0.249 + 0.601i)6-s + (−0.923 + 0.382i)7-s + (1.19 + 1.19i)8-s + (0.614 + 0.614i)9-s + (−0.182 + 0.0757i)10-s + (0.964 + 2.32i)11-s + (−2.42 − 1.00i)12-s − 6.23i·13-s + (−0.170 + 0.411i)14-s + (0.458 − 0.458i)15-s − 2.84·16-s + (4.09 + 0.449i)17-s + ⋯
L(s)  = 1  + (0.222 − 0.222i)2-s + (−0.322 + 0.778i)3-s + 0.900i·4-s + (−0.183 − 0.0760i)5-s + (0.101 + 0.245i)6-s + (−0.349 + 0.144i)7-s + (0.423 + 0.423i)8-s + (0.204 + 0.204i)9-s + (−0.0578 + 0.0239i)10-s + (0.290 + 0.702i)11-s + (−0.701 − 0.290i)12-s − 1.72i·13-s + (−0.0456 + 0.110i)14-s + (0.118 − 0.118i)15-s − 0.711·16-s + (0.994 + 0.109i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 119 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.413 - 0.910i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 119 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.413 - 0.910i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(119\)    =    \(7 \cdot 17\)
Sign: $0.413 - 0.910i$
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: \((2,\ 119,\ (\ :1/2),\ 0.413 - 0.910i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.884720 + 0.570099i\)
\(L(\frac12)\) \(\approx\) \(0.884720 + 0.570099i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-4.09 - 0.449i)T \)
good2 \( 1 + (-0.315 + 0.315i)T - 2iT^{2} \)
3 \( 1 + (0.558 - 1.34i)T + (-2.12 - 2.12i)T^{2} \)
5 \( 1 + (0.410 + 0.169i)T + (3.53 + 3.53i)T^{2} \)
11 \( 1 + (-0.964 - 2.32i)T + (-7.77 + 7.77i)T^{2} \)
13 \( 1 + 6.23iT - 13T^{2} \)
19 \( 1 + (-3.24 + 3.24i)T - 19iT^{2} \)
23 \( 1 + (0.682 + 1.64i)T + (-16.2 + 16.2i)T^{2} \)
29 \( 1 + (-3.85 - 1.59i)T + (20.5 + 20.5i)T^{2} \)
31 \( 1 + (0.141 - 0.342i)T + (-21.9 - 21.9i)T^{2} \)
37 \( 1 + (-1.98 + 4.79i)T + (-26.1 - 26.1i)T^{2} \)
41 \( 1 + (8.07 - 3.34i)T + (28.9 - 28.9i)T^{2} \)
43 \( 1 + (-6.85 - 6.85i)T + 43iT^{2} \)
47 \( 1 + 1.55iT - 47T^{2} \)
53 \( 1 + (-8.32 + 8.32i)T - 53iT^{2} \)
59 \( 1 + (-2.45 - 2.45i)T + 59iT^{2} \)
61 \( 1 + (4.68 - 1.93i)T + (43.1 - 43.1i)T^{2} \)
67 \( 1 + 4.54T + 67T^{2} \)
71 \( 1 + (4.23 - 10.2i)T + (-50.2 - 50.2i)T^{2} \)
73 \( 1 + (0.476 + 0.197i)T + (51.6 + 51.6i)T^{2} \)
79 \( 1 + (4.79 + 11.5i)T + (-55.8 + 55.8i)T^{2} \)
83 \( 1 + (4.38 - 4.38i)T - 83iT^{2} \)
89 \( 1 + 2.42iT - 89T^{2} \)
97 \( 1 + (8.05 + 3.33i)T + (68.5 + 68.5i)T^{2} \)
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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

−13.40584723414695854670821590621, −12.58070499010614539179524986979, −11.75797741332923456825352868841, −10.53080208985327181375262828929, −9.724655657153549541898993087525, −8.249321939071013573375475107987, −7.28883488035388636016418094377, −5.45203319706581600549641971025, −4.28504105302997972393160286076, −3.00431864979084458660471315720, 1.38652574888595104926993612952, 3.94172631706400351351344472566, 5.66383303755689227759932838498, 6.55999009612214049008362540582, 7.47082263778547578983691241067, 9.209579811211567364136796797413, 10.11398887123763751676987884251, 11.52836852675937773656607933266, 12.14796883046272317363676802472, 13.68387432035713963663499414152

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