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Properties

Label 2-119-119.104-c4-0-0
Degree $2$
Conductor $119$
Sign $0.650 + 0.759i$
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  + (−4.63 − 4.63i)2-s + (−5.03 − 12.1i)3-s + 26.8i·4-s + (−36.7 + 15.2i)5-s + (−32.9 + 79.6i)6-s + (−36.3 − 32.8i)7-s + (50.4 − 50.4i)8-s + (−65.1 + 65.1i)9-s + (240. + 99.8i)10-s + (−37.0 − 15.3i)11-s + (326. − 135. i)12-s − 90.8·13-s + (16.2 + 320. i)14-s + (370. + 370. i)15-s − 37.0·16-s + (23.6 − 288. i)17-s + ⋯
L(s)  = 1  + (−1.15 − 1.15i)2-s + (−0.559 − 1.35i)3-s + 1.68i·4-s + (−1.47 + 0.609i)5-s + (−0.915 + 2.21i)6-s + (−0.741 − 0.670i)7-s + (0.788 − 0.788i)8-s + (−0.804 + 0.804i)9-s + (2.40 + 0.998i)10-s + (−0.306 − 0.126i)11-s + (2.27 − 0.940i)12-s − 0.537·13-s + (0.0827 + 1.63i)14-s + (1.64 + 1.64i)15-s − 0.144·16-s + (0.0817 − 0.996i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 119 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.650 + 0.759i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 119 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.650 + 0.759i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(119\)    =    \(7 \cdot 17\)
Sign: $0.650 + 0.759i$
Analytic conductor: \(12.3010\)
Root analytic conductor: \(3.50728\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{119} (104, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 119,\ (\ :2),\ 0.650 + 0.759i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.0390724 - 0.0179952i\)
\(L(\frac12)\) \(\approx\) \(0.0390724 - 0.0179952i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (36.3 + 32.8i)T \)
17 \( 1 + (-23.6 + 288. i)T \)
good2 \( 1 + (4.63 + 4.63i)T + 16iT^{2} \)
3 \( 1 + (5.03 + 12.1i)T + (-57.2 + 57.2i)T^{2} \)
5 \( 1 + (36.7 - 15.2i)T + (441. - 441. i)T^{2} \)
11 \( 1 + (37.0 + 15.3i)T + (1.03e4 + 1.03e4i)T^{2} \)
13 \( 1 + 90.8T + 2.85e4T^{2} \)
19 \( 1 + (234. - 234. i)T - 1.30e5iT^{2} \)
23 \( 1 + (877. + 363. i)T + (1.97e5 + 1.97e5i)T^{2} \)
29 \( 1 + (-107. - 259. i)T + (-5.00e5 + 5.00e5i)T^{2} \)
31 \( 1 + (421. + 1.01e3i)T + (-6.53e5 + 6.53e5i)T^{2} \)
37 \( 1 + (-1.93e3 + 799. i)T + (1.32e6 - 1.32e6i)T^{2} \)
41 \( 1 + (377. + 156. i)T + (1.99e6 + 1.99e6i)T^{2} \)
43 \( 1 + (546. - 546. i)T - 3.41e6iT^{2} \)
47 \( 1 + 763.T + 4.87e6T^{2} \)
53 \( 1 + (468. + 468. i)T + 7.89e6iT^{2} \)
59 \( 1 + (4.27e3 + 4.27e3i)T + 1.21e7iT^{2} \)
61 \( 1 + (-4.67e3 - 1.93e3i)T + (9.79e6 + 9.79e6i)T^{2} \)
67 \( 1 - 5.55e3T + 2.01e7T^{2} \)
71 \( 1 + (2.70e3 - 1.11e3i)T + (1.79e7 - 1.79e7i)T^{2} \)
73 \( 1 + (-5.47e3 + 2.26e3i)T + (2.00e7 - 2.00e7i)T^{2} \)
79 \( 1 + (-172. - 71.2i)T + (2.75e7 + 2.75e7i)T^{2} \)
83 \( 1 + (-1.07e3 + 1.07e3i)T - 4.74e7iT^{2} \)
89 \( 1 - 1.11e4T + 6.27e7T^{2} \)
97 \( 1 + (-3.09e3 + 1.28e3i)T + (6.25e7 - 6.25e7i)T^{2} \)
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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

−12.24901912831820515229236998638, −11.59762302933463832052019612996, −10.78528458098960992162638684996, −9.736643858505764537963607856656, −7.991857249372951957160215193413, −7.59964912369846780830472547338, −6.45765135150150823614387513966, −3.81079143312198438174636598356, −2.40140780933885535049378596013, −0.53575643847658641724002097621, 0.06215825686994333630263539884, 3.84052880503324417400646600839, 5.09617690294076179727491213484, 6.29007482823584608565987083916, 7.76194524701268348950931323254, 8.626923885164325094624002350491, 9.577611956526747376270649197870, 10.40960101630553320943087085573, 11.64758727541669781656157098386, 12.65197519326771303100828567504

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