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Properties

Label 2-119-7.6-c4-0-10
Degree $2$
Conductor $119$
Sign $-0.886 + 0.463i$
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  + 0.719·2-s + 15.8i·3-s − 15.4·4-s + 29.0i·5-s + 11.3i·6-s + (22.7 + 43.4i)7-s − 22.6·8-s − 168.·9-s + 20.8i·10-s + 143.·11-s − 244. i·12-s + 69.9i·13-s + (16.3 + 31.2i)14-s − 458.·15-s + 231.·16-s + 70.0i·17-s + ⋯
L(s)  = 1  + 0.179·2-s + 1.75i·3-s − 0.967·4-s + 1.16i·5-s + 0.315i·6-s + (0.463 + 0.886i)7-s − 0.353·8-s − 2.08·9-s + 0.208i·10-s + 1.18·11-s − 1.69i·12-s + 0.413i·13-s + (0.0833 + 0.159i)14-s − 2.03·15-s + 0.903·16-s + 0.242i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 119 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.886 + 0.463i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 119 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.886 + 0.463i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(119\)    =    \(7 \cdot 17\)
Sign: $-0.886 + 0.463i$
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: \((2,\ 119,\ (\ :2),\ -0.886 + 0.463i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.319394 - 1.29991i\)
\(L(\frac12)\) \(\approx\) \(0.319394 - 1.29991i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-22.7 - 43.4i)T \)
17 \( 1 - 70.0iT \)
good2 \( 1 - 0.719T + 16T^{2} \)
3 \( 1 - 15.8iT - 81T^{2} \)
5 \( 1 - 29.0iT - 625T^{2} \)
11 \( 1 - 143.T + 1.46e4T^{2} \)
13 \( 1 - 69.9iT - 2.85e4T^{2} \)
19 \( 1 + 504. iT - 1.30e5T^{2} \)
23 \( 1 + 374.T + 2.79e5T^{2} \)
29 \( 1 - 1.08e3T + 7.07e5T^{2} \)
31 \( 1 - 719. iT - 9.23e5T^{2} \)
37 \( 1 + 93.4T + 1.87e6T^{2} \)
41 \( 1 + 1.10e3iT - 2.82e6T^{2} \)
43 \( 1 + 2.62e3T + 3.41e6T^{2} \)
47 \( 1 - 3.62e3iT - 4.87e6T^{2} \)
53 \( 1 - 2.60e3T + 7.89e6T^{2} \)
59 \( 1 + 2.46e3iT - 1.21e7T^{2} \)
61 \( 1 - 4.79e3iT - 1.38e7T^{2} \)
67 \( 1 - 7.03e3T + 2.01e7T^{2} \)
71 \( 1 - 1.81e3T + 2.54e7T^{2} \)
73 \( 1 + 5.16e3iT - 2.83e7T^{2} \)
79 \( 1 + 3.53e3T + 3.89e7T^{2} \)
83 \( 1 + 9.59e3iT - 4.74e7T^{2} \)
89 \( 1 - 8.49e3iT - 6.27e7T^{2} \)
97 \( 1 + 1.29e4iT - 8.85e7T^{2} \)
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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

−13.91863303740211481794153879425, −12.03830890401631213227177475038, −11.15421494599779956324408510978, −10.15182266301552199414669234446, −9.221902566144697901696409628992, −8.562959586933219623406349413617, −6.42346615524888421622858852032, −5.09129575809741804129153210620, −4.14039596663794838529726392567, −2.98439024037647912605816708605, 0.62244090988119521457291422924, 1.45389636811814463026147850981, 3.94495420650703647128045475591, 5.34275880349747415578431030576, 6.62927048385634071733604623286, 8.052175582728597442904952582282, 8.469888211608606017367353291180, 9.889988090367946954781837043462, 11.77311010271389176493168400685, 12.36167140248600273556055027163

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