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Properties

Label 2-304-19.5-c1-0-3
Degree $2$
Conductor $304$
Sign $0.992 - 0.120i$
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  + (0.0603 − 0.342i)3-s + (1.17 + 0.984i)5-s + (−0.152 − 0.264i)7-s + (2.70 + 0.984i)9-s + (0.152 − 0.264i)11-s + (0.0603 + 0.342i)13-s + (0.407 − 0.342i)15-s + (1.76 − 0.642i)17-s + (3.06 + 3.10i)19-s + (−0.0996 + 0.0362i)21-s + (2.17 − 1.82i)23-s + (−0.460 − 2.61i)25-s + (1.02 − 1.76i)27-s + (−3.29 − 1.20i)29-s + (2.91 + 5.04i)31-s + ⋯
L(s)  = 1  + (0.0348 − 0.197i)3-s + (0.524 + 0.440i)5-s + (−0.0577 − 0.0999i)7-s + (0.901 + 0.328i)9-s + (0.0460 − 0.0797i)11-s + (0.0167 + 0.0948i)13-s + (0.105 − 0.0883i)15-s + (0.428 − 0.155i)17-s + (0.702 + 0.711i)19-s + (−0.0217 + 0.00791i)21-s + (0.453 − 0.380i)23-s + (−0.0921 − 0.522i)25-s + (0.196 − 0.340i)27-s + (−0.612 − 0.222i)29-s + (0.522 + 0.905i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 - 0.120i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.992 - 0.120i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(304\)    =    \(2^{4} \cdot 19\)
Sign: $0.992 - 0.120i$
Analytic conductor: \(2.42745\)
Root analytic conductor: \(1.55802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{304} (81, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 304,\ (\ :1/2),\ 0.992 - 0.120i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.50152 + 0.0905061i\)
\(L(\frac12)\) \(\approx\) \(1.50152 + 0.0905061i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.06 - 3.10i)T \)
good3 \( 1 + (-0.0603 + 0.342i)T + (-2.81 - 1.02i)T^{2} \)
5 \( 1 + (-1.17 - 0.984i)T + (0.868 + 4.92i)T^{2} \)
7 \( 1 + (0.152 + 0.264i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.152 + 0.264i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.0603 - 0.342i)T + (-12.2 + 4.44i)T^{2} \)
17 \( 1 + (-1.76 + 0.642i)T + (13.0 - 10.9i)T^{2} \)
23 \( 1 + (-2.17 + 1.82i)T + (3.99 - 22.6i)T^{2} \)
29 \( 1 + (3.29 + 1.20i)T + (22.2 + 18.6i)T^{2} \)
31 \( 1 + (-2.91 - 5.04i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 8.12T + 37T^{2} \)
41 \( 1 + (-0.429 + 2.43i)T + (-38.5 - 14.0i)T^{2} \)
43 \( 1 + (-0.890 - 0.747i)T + (7.46 + 42.3i)T^{2} \)
47 \( 1 + (5.87 + 2.13i)T + (36.0 + 30.2i)T^{2} \)
53 \( 1 + (5.52 - 4.63i)T + (9.20 - 52.1i)T^{2} \)
59 \( 1 + (10.7 - 3.89i)T + (45.1 - 37.9i)T^{2} \)
61 \( 1 + (8.71 - 7.31i)T + (10.5 - 60.0i)T^{2} \)
67 \( 1 + (-0.340 - 0.123i)T + (51.3 + 43.0i)T^{2} \)
71 \( 1 + (9.93 + 8.33i)T + (12.3 + 69.9i)T^{2} \)
73 \( 1 + (-1.04 + 5.90i)T + (-68.5 - 24.9i)T^{2} \)
79 \( 1 + (1.57 - 8.90i)T + (-74.2 - 27.0i)T^{2} \)
83 \( 1 + (8.28 + 14.3i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (0.180 + 1.02i)T + (-83.6 + 30.4i)T^{2} \)
97 \( 1 + (-8.83 + 3.21i)T + (74.3 - 62.3i)T^{2} \)
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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

−11.84850045986803596078151277840, −10.52924615902323195095079158802, −10.09530991456910056492002726888, −8.977367416289742430554500566277, −7.74300268967570804459832087205, −6.92572723519106247038329095613, −5.87940460290630132760276483622, −4.63425334117247911339896243327, −3.19733476705968351236261183651, −1.65918249475121077308990668986, 1.48114941818230087355862444908, 3.28632411852099735555284537665, 4.64027863500454777428445074893, 5.63069881013247044758825668803, 6.84686330803034442430973191830, 7.83706774003531945249800120850, 9.219146271603213903674406145816, 9.602995879851952192221205376165, 10.68804739719365953771968193993, 11.74033420609949310462492512777

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