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Properties

Label 2-119-7.6-c4-0-33
Degree $2$
Conductor $119$
Sign $0.952 + 0.304i$
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  + 6.58·2-s − 5.18i·3-s + 27.4·4-s + 26.0i·5-s − 34.1i·6-s + (14.9 − 46.6i)7-s + 75.2·8-s + 54.0·9-s + 171. i·10-s + 159.·11-s − 142. i·12-s − 89.9i·13-s + (98.4 − 307. i)14-s + 135.·15-s + 56.9·16-s + 70.0i·17-s + ⋯
L(s)  = 1  + 1.64·2-s − 0.576i·3-s + 1.71·4-s + 1.04i·5-s − 0.949i·6-s + (0.304 − 0.952i)7-s + 1.17·8-s + 0.667·9-s + 1.71i·10-s + 1.32·11-s − 0.987i·12-s − 0.532i·13-s + (0.502 − 1.56i)14-s + 0.600·15-s + 0.222·16-s + 0.242i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 119 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.952 + 0.304i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 119 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.952 + 0.304i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(119\)    =    \(7 \cdot 17\)
Sign: $0.952 + 0.304i$
Analytic conductor: \(12.3010\)
Root analytic conductor: \(3.50728\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{119} (69, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 119,\ (\ :2),\ 0.952 + 0.304i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(4.69002 - 0.732202i\)
\(L(\frac12)\) \(\approx\) \(4.69002 - 0.732202i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-14.9 + 46.6i)T \)
17 \( 1 - 70.0iT \)
good2 \( 1 - 6.58T + 16T^{2} \)
3 \( 1 + 5.18iT - 81T^{2} \)
5 \( 1 - 26.0iT - 625T^{2} \)
11 \( 1 - 159.T + 1.46e4T^{2} \)
13 \( 1 + 89.9iT - 2.85e4T^{2} \)
19 \( 1 - 272. iT - 1.30e5T^{2} \)
23 \( 1 + 918.T + 2.79e5T^{2} \)
29 \( 1 + 403.T + 7.07e5T^{2} \)
31 \( 1 - 166. iT - 9.23e5T^{2} \)
37 \( 1 + 1.01e3T + 1.87e6T^{2} \)
41 \( 1 - 1.43e3iT - 2.82e6T^{2} \)
43 \( 1 - 256.T + 3.41e6T^{2} \)
47 \( 1 - 2.00e3iT - 4.87e6T^{2} \)
53 \( 1 + 2.28e3T + 7.89e6T^{2} \)
59 \( 1 - 5.11e3iT - 1.21e7T^{2} \)
61 \( 1 + 4.87e3iT - 1.38e7T^{2} \)
67 \( 1 - 4.68e3T + 2.01e7T^{2} \)
71 \( 1 - 4.25e3T + 2.54e7T^{2} \)
73 \( 1 + 8.79e3iT - 2.83e7T^{2} \)
79 \( 1 + 1.11e4T + 3.89e7T^{2} \)
83 \( 1 - 1.22e3iT - 4.74e7T^{2} \)
89 \( 1 - 5.47e3iT - 6.27e7T^{2} \)
97 \( 1 + 1.49e4iT - 8.85e7T^{2} \)
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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

−12.84334855485919662837065094511, −12.02024417591417874899392647662, −11.02780649057888074904603236766, −10.00546747844236650399698066389, −7.77212435556028621418017139439, −6.81363028555487820823185948633, −6.10045364896192073818239587745, −4.32118928988985495719505683909, −3.46956691858426145249355971534, −1.71328429673370313977103550802, 1.88670833750943628937093081523, 3.83524477501335682654001597985, 4.63315059808396724563887747389, 5.59140540714752829687551251982, 6.83486778615142293255903018595, 8.709670356647570867385383762673, 9.596313513751922960886579608206, 11.34555398795064595450915827370, 12.08933158272973142138333043019, 12.75068210002753975138312355260

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