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Properties

Label 2-304-19.9-c1-0-2
Degree $2$
Conductor $304$
Sign $0.524 - 0.851i$
Analytic cond. $2.42745$
Root an. cond. $1.55802$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.80 + 1.51i)3-s + (2.20 + 0.801i)5-s + (2.07 − 3.59i)7-s + (0.447 − 2.53i)9-s + (2.68 + 4.64i)11-s + (−2.26 − 1.89i)13-s + (−5.19 + 1.89i)15-s + (0.962 + 5.46i)17-s + (2.01 + 3.86i)19-s + (1.70 + 9.66i)21-s + (1.68 − 0.613i)23-s + (0.374 + 0.314i)25-s + (−0.497 − 0.862i)27-s + (−0.0374 + 0.212i)29-s + (−1.20 + 2.08i)31-s + ⋯
L(s)  = 1  + (−1.04 + 0.876i)3-s + (0.984 + 0.358i)5-s + (0.785 − 1.36i)7-s + (0.149 − 0.846i)9-s + (0.809 + 1.40i)11-s + (−0.627 − 0.526i)13-s + (−1.34 + 0.488i)15-s + (0.233 + 1.32i)17-s + (0.461 + 0.887i)19-s + (0.371 + 2.10i)21-s + (0.351 − 0.127i)23-s + (0.0749 + 0.0628i)25-s + (−0.0958 − 0.165i)27-s + (−0.00695 + 0.0394i)29-s + (−0.216 + 0.374i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.524 - 0.851i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.524 - 0.851i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(304\)    =    \(2^{4} \cdot 19\)
Sign: $0.524 - 0.851i$
Analytic conductor: \(2.42745\)
Root analytic conductor: \(1.55802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{304} (161, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 304,\ (\ :1/2),\ 0.524 - 0.851i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.03532 + 0.578265i\)
\(L(\frac12)\) \(\approx\) \(1.03532 + 0.578265i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
19 \( 1 + (-2.01 - 3.86i)T \)
good3 \( 1 + (1.80 - 1.51i)T + (0.520 - 2.95i)T^{2} \)
5 \( 1 + (-2.20 - 0.801i)T + (3.83 + 3.21i)T^{2} \)
7 \( 1 + (-2.07 + 3.59i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-2.68 - 4.64i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (2.26 + 1.89i)T + (2.25 + 12.8i)T^{2} \)
17 \( 1 + (-0.962 - 5.46i)T + (-15.9 + 5.81i)T^{2} \)
23 \( 1 + (-1.68 + 0.613i)T + (17.6 - 14.7i)T^{2} \)
29 \( 1 + (0.0374 - 0.212i)T + (-27.2 - 9.91i)T^{2} \)
31 \( 1 + (1.20 - 2.08i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 6.54T + 37T^{2} \)
41 \( 1 + (1.33 - 1.12i)T + (7.11 - 40.3i)T^{2} \)
43 \( 1 + (1.55 + 0.564i)T + (32.9 + 27.6i)T^{2} \)
47 \( 1 + (-1.19 + 6.75i)T + (-44.1 - 16.0i)T^{2} \)
53 \( 1 + (-5.94 + 2.16i)T + (40.6 - 34.0i)T^{2} \)
59 \( 1 + (1.77 + 10.0i)T + (-55.4 + 20.1i)T^{2} \)
61 \( 1 + (9.39 - 3.41i)T + (46.7 - 39.2i)T^{2} \)
67 \( 1 + (1.55 - 8.81i)T + (-62.9 - 22.9i)T^{2} \)
71 \( 1 + (10.0 + 3.67i)T + (54.3 + 45.6i)T^{2} \)
73 \( 1 + (-9.55 + 8.01i)T + (12.6 - 71.8i)T^{2} \)
79 \( 1 + (-4.29 + 3.60i)T + (13.7 - 77.7i)T^{2} \)
83 \( 1 + (3.65 - 6.33i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-4.15 - 3.48i)T + (15.4 + 87.6i)T^{2} \)
97 \( 1 + (3.09 + 17.5i)T + (-91.1 + 33.1i)T^{2} \)
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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

−11.69299954173235484313255275699, −10.58558243649901299409321724781, −10.24780458162128732228860945411, −9.601190674783776815105787113117, −7.88836747414438599105954641413, −6.85876879159669436585439174216, −5.79682513562112546783898298555, −4.77484355376381875408246041935, −3.93967075889178631769919082944, −1.65732909555196299221679404999, 1.17726516796047496342885496411, 2.57665058915126704731403463184, 4.98852195466373302195681213200, 5.66156769119175159242910324718, 6.37834413634304671576355072553, 7.55776409100361043602718422361, 8.995321528476939688515636294255, 9.366918926922331606360485256306, 11.13955387122886637935812664211, 11.64675695611520796045575405816

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