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Properties

Label 2-119-119.101-c4-0-31
Degree $2$
Conductor $119$
Sign $0.0306 + 0.999i$
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  + (0.362 + 0.627i)2-s + (−7.31 + 12.6i)3-s + (7.73 − 13.4i)4-s + (−4.02 − 6.97i)5-s − 10.5·6-s + (−10.9 + 47.7i)7-s + 22.8·8-s + (−66.5 − 115. i)9-s + (2.91 − 5.05i)10-s + (−176. − 101. i)11-s + (113. + 196. i)12-s + 50.3i·13-s + (−33.9 + 10.4i)14-s + 117.·15-s + (−115. − 200. i)16-s + (110. − 266. i)17-s + ⋯
L(s)  = 1  + (0.0905 + 0.156i)2-s + (−0.812 + 1.40i)3-s + (0.483 − 0.837i)4-s + (−0.161 − 0.279i)5-s − 0.294·6-s + (−0.223 + 0.974i)7-s + 0.356·8-s + (−0.821 − 1.42i)9-s + (0.0291 − 0.0505i)10-s + (−1.45 − 0.840i)11-s + (0.786 + 1.36i)12-s + 0.297i·13-s + (−0.173 + 0.0531i)14-s + 0.523·15-s + (−0.451 − 0.781i)16-s + (0.383 − 0.923i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 119 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0306 + 0.999i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 119 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.0306 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(119\)    =    \(7 \cdot 17\)
Sign: $0.0306 + 0.999i$
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: \((2,\ 119,\ (\ :2),\ 0.0306 + 0.999i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.374409 - 0.363111i\)
\(L(\frac12)\) \(\approx\) \(0.374409 - 0.363111i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (10.9 - 47.7i)T \)
17 \( 1 + (-110. + 266. i)T \)
good2 \( 1 + (-0.362 - 0.627i)T + (-8 + 13.8i)T^{2} \)
3 \( 1 + (7.31 - 12.6i)T + (-40.5 - 70.1i)T^{2} \)
5 \( 1 + (4.02 + 6.97i)T + (-312.5 + 541. i)T^{2} \)
11 \( 1 + (176. + 101. i)T + (7.32e3 + 1.26e4i)T^{2} \)
13 \( 1 - 50.3iT - 2.85e4T^{2} \)
19 \( 1 + (111. - 64.5i)T + (6.51e4 - 1.12e5i)T^{2} \)
23 \( 1 + (-676. + 390. i)T + (1.39e5 - 2.42e5i)T^{2} \)
29 \( 1 + 667. iT - 7.07e5T^{2} \)
31 \( 1 + (-138. + 239. i)T + (-4.61e5 - 7.99e5i)T^{2} \)
37 \( 1 + (1.82e3 - 1.05e3i)T + (9.37e5 - 1.62e6i)T^{2} \)
41 \( 1 + 1.60e3T + 2.82e6T^{2} \)
43 \( 1 + 2.46e3T + 3.41e6T^{2} \)
47 \( 1 + (-772. + 446. i)T + (2.43e6 - 4.22e6i)T^{2} \)
53 \( 1 + (1.68e3 - 2.91e3i)T + (-3.94e6 - 6.83e6i)T^{2} \)
59 \( 1 + (-1.58e3 - 914. i)T + (6.05e6 + 1.04e7i)T^{2} \)
61 \( 1 + (1.83e3 + 3.18e3i)T + (-6.92e6 + 1.19e7i)T^{2} \)
67 \( 1 + (-1.73e3 + 2.99e3i)T + (-1.00e7 - 1.74e7i)T^{2} \)
71 \( 1 - 3.82e3iT - 2.54e7T^{2} \)
73 \( 1 + (-2.56e3 + 4.43e3i)T + (-1.41e7 - 2.45e7i)T^{2} \)
79 \( 1 + (4.26e3 - 2.46e3i)T + (1.94e7 - 3.37e7i)T^{2} \)
83 \( 1 - 1.19e4iT - 4.74e7T^{2} \)
89 \( 1 + (-7.38e3 + 4.26e3i)T + (3.13e7 - 5.43e7i)T^{2} \)
97 \( 1 + 1.49e4T + 8.85e7T^{2} \)
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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

−12.19593351097534495527733475864, −11.25170661214822491690297296455, −10.48836368671572461055909617586, −9.652185242407968700555875471681, −8.465519075384078471586870639477, −6.50190297031230779399950967706, −5.37878150557341573405570984674, −4.89726375428803070459244247155, −2.86358149014132141316326870807, −0.23002126992219746681427933638, 1.64399440363472538094006361679, 3.24618543198817364350668137569, 5.19879255331367413529027215509, 6.88448005478289101363080366753, 7.24412866271633663943314680891, 8.186464860794175291938438796351, 10.45031924248342424841119654398, 11.06003210418252300725896320266, 12.23795449745400611361637314672, 12.99284449464709246841868423852

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