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Properties

Label 2-592-37.36-c3-0-49
Degree $2$
Conductor $592$
Sign $-0.343 + 0.939i$
Analytic cond. $34.9291$
Root an. cond. $5.91008$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 8.15·3-s − 6.18i·5-s − 23.1·7-s + 39.5·9-s − 13.1·11-s − 30.8i·13-s − 50.4i·15-s − 87.4i·17-s + 93.7i·19-s − 188.·21-s − 73.1i·23-s + 86.7·25-s + 102.·27-s − 199. i·29-s − 125. i·31-s + ⋯
L(s)  = 1  + 1.56·3-s − 0.553i·5-s − 1.25·7-s + 1.46·9-s − 0.361·11-s − 0.657i·13-s − 0.868i·15-s − 1.24i·17-s + 1.13i·19-s − 1.96·21-s − 0.663i·23-s + 0.694·25-s + 0.727·27-s − 1.27i·29-s − 0.726i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.343 + 0.939i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 592 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.343 + 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(592\)    =    \(2^{4} \cdot 37\)
Sign: $-0.343 + 0.939i$
Analytic conductor: \(34.9291\)
Root analytic conductor: \(5.91008\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{592} (369, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 592,\ (\ :3/2),\ -0.343 + 0.939i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.215795210\)
\(L(\frac12)\) \(\approx\) \(2.215795210\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
37 \( 1 + (-77.2 + 211. i)T \)
good3 \( 1 - 8.15T + 27T^{2} \)
5 \( 1 + 6.18iT - 125T^{2} \)
7 \( 1 + 23.1T + 343T^{2} \)
11 \( 1 + 13.1T + 1.33e3T^{2} \)
13 \( 1 + 30.8iT - 2.19e3T^{2} \)
17 \( 1 + 87.4iT - 4.91e3T^{2} \)
19 \( 1 - 93.7iT - 6.85e3T^{2} \)
23 \( 1 + 73.1iT - 1.21e4T^{2} \)
29 \( 1 + 199. iT - 2.43e4T^{2} \)
31 \( 1 + 125. iT - 2.97e4T^{2} \)
41 \( 1 + 339.T + 6.89e4T^{2} \)
43 \( 1 + 501. iT - 7.95e4T^{2} \)
47 \( 1 + 282.T + 1.03e5T^{2} \)
53 \( 1 + 205.T + 1.48e5T^{2} \)
59 \( 1 - 680. iT - 2.05e5T^{2} \)
61 \( 1 + 352. iT - 2.26e5T^{2} \)
67 \( 1 + 739.T + 3.00e5T^{2} \)
71 \( 1 - 188.T + 3.57e5T^{2} \)
73 \( 1 - 1.21e3T + 3.89e5T^{2} \)
79 \( 1 - 750. iT - 4.93e5T^{2} \)
83 \( 1 - 982.T + 5.71e5T^{2} \)
89 \( 1 - 1.14e3iT - 7.04e5T^{2} \)
97 \( 1 - 273. iT - 9.12e5T^{2} \)
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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

−9.722966870627857147121501611460, −9.192474296331193128915234784517, −8.283080429778572001000588542883, −7.62582372364194777207116022418, −6.55569968536196219060775707267, −5.32094781783631414071250636225, −3.98594717128080631676940363817, −3.12213199038235324040871865542, −2.26308018514780015700883272982, −0.49308294236575830476138944574, 1.73120404557009029130890169481, 3.09439441659943953221353199570, 3.31087275998987409710309798927, 4.73294104136194157030255385048, 6.41853172415281267221124892547, 6.95973528162052359024061968994, 8.035693245529174641295921042239, 8.861216170327524264391990871655, 9.533415940855918493064731756778, 10.28122486873779498585230198404

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