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Properties

Label 2-119-119.101-c4-0-14
Degree $2$
Conductor $119$
Sign $0.237 - 0.971i$
Analytic cond. $12.3010$
Root an. cond. $3.50728$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.07 + 1.85i)2-s + (−3.88 + 6.73i)3-s + (5.69 − 9.86i)4-s + (−18.2 − 31.6i)5-s − 16.7·6-s + (−42.5 + 24.3i)7-s + 58.8·8-s + (10.2 + 17.7i)9-s + (39.2 − 68.0i)10-s + (188. + 108. i)11-s + (44.2 + 76.7i)12-s + 210. i·13-s + (−90.9 − 52.8i)14-s + 284.·15-s + (−27.9 − 48.3i)16-s + (45.8 + 285. i)17-s + ⋯
L(s)  = 1  + (0.268 + 0.464i)2-s + (−0.432 + 0.748i)3-s + (0.355 − 0.616i)4-s + (−0.731 − 1.26i)5-s − 0.464·6-s + (−0.867 + 0.497i)7-s + 0.919·8-s + (0.126 + 0.219i)9-s + (0.392 − 0.680i)10-s + (1.55 + 0.898i)11-s + (0.307 + 0.532i)12-s + 1.24i·13-s + (−0.464 − 0.269i)14-s + 1.26·15-s + (−0.109 − 0.189i)16-s + (0.158 + 0.987i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 119 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.237 - 0.971i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 119 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.237 - 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(119\)    =    \(7 \cdot 17\)
Sign: $0.237 - 0.971i$
Analytic conductor: \(12.3010\)
Root analytic conductor: \(3.50728\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{119} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 119,\ (\ :2),\ 0.237 - 0.971i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(1.32894 + 1.04262i\)
\(L(\frac12)\) \(\approx\) \(1.32894 + 1.04262i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (42.5 - 24.3i)T \)
17 \( 1 + (-45.8 - 285. i)T \)
good2 \( 1 + (-1.07 - 1.85i)T + (-8 + 13.8i)T^{2} \)
3 \( 1 + (3.88 - 6.73i)T + (-40.5 - 70.1i)T^{2} \)
5 \( 1 + (18.2 + 31.6i)T + (-312.5 + 541. i)T^{2} \)
11 \( 1 + (-188. - 108. i)T + (7.32e3 + 1.26e4i)T^{2} \)
13 \( 1 - 210. iT - 2.85e4T^{2} \)
19 \( 1 + (-348. + 201. i)T + (6.51e4 - 1.12e5i)T^{2} \)
23 \( 1 + (-458. + 264. i)T + (1.39e5 - 2.42e5i)T^{2} \)
29 \( 1 - 236. iT - 7.07e5T^{2} \)
31 \( 1 + (235. - 408. i)T + (-4.61e5 - 7.99e5i)T^{2} \)
37 \( 1 + (68.1 - 39.3i)T + (9.37e5 - 1.62e6i)T^{2} \)
41 \( 1 - 2.49e3T + 2.82e6T^{2} \)
43 \( 1 - 1.67e3T + 3.41e6T^{2} \)
47 \( 1 + (2.95e3 - 1.70e3i)T + (2.43e6 - 4.22e6i)T^{2} \)
53 \( 1 + (1.95e3 - 3.38e3i)T + (-3.94e6 - 6.83e6i)T^{2} \)
59 \( 1 + (3.49e3 + 2.01e3i)T + (6.05e6 + 1.04e7i)T^{2} \)
61 \( 1 + (-1.25e3 - 2.16e3i)T + (-6.92e6 + 1.19e7i)T^{2} \)
67 \( 1 + (2.18e3 - 3.78e3i)T + (-1.00e7 - 1.74e7i)T^{2} \)
71 \( 1 - 2.32e3iT - 2.54e7T^{2} \)
73 \( 1 + (-2.89e3 + 5.00e3i)T + (-1.41e7 - 2.45e7i)T^{2} \)
79 \( 1 + (664. - 383. i)T + (1.94e7 - 3.37e7i)T^{2} \)
83 \( 1 + 1.58e3iT - 4.74e7T^{2} \)
89 \( 1 + (-215. + 124. i)T + (3.13e7 - 5.43e7i)T^{2} \)
97 \( 1 - 5.00e3T + 8.85e7T^{2} \)
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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

−12.85249367146940933424722934844, −11.98749743536990155341739078297, −11.03039471416870174884450450278, −9.596304904586223654590735983666, −9.074661445215613730214170717157, −7.27461524316210713348099741322, −6.16675836043385096191326268041, −4.83683602190666702000490682628, −4.17202675020428360830060979430, −1.38931471257944332122532830043, 0.817955439651870121892259860050, 3.15906966586644220893153718112, 3.64058307040135578077240368967, 6.18187508790106735071599072546, 7.08194025422475795290481827194, 7.68932829848989825005934936403, 9.611649808888933350842455756833, 11.00121895586634083460045838102, 11.55100970131628385240997633199, 12.38973291290881576867686101486

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