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Properties

Label 2-592-37.3-c1-0-5
Degree $2$
Conductor $592$
Sign $0.874 - 0.484i$
Analytic cond. $4.72714$
Root an. cond. $2.17419$
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  + (−1.43 + 1.20i)3-s + (0.273 − 0.751i)5-s + (0.138 + 0.0503i)7-s + (0.0923 − 0.524i)9-s + (2.40 − 4.16i)11-s + (1.91 − 0.338i)13-s + (0.514 + 1.41i)15-s + (−3.43 − 0.606i)17-s + (4.33 + 5.16i)19-s + (−0.260 + 0.0947i)21-s + (3.61 − 2.08i)23-s + (3.33 + 2.80i)25-s + (−2.31 − 4.01i)27-s + (2.63 + 1.51i)29-s + 8.13i·31-s + ⋯
L(s)  = 1  + (−0.831 + 0.697i)3-s + (0.122 − 0.336i)5-s + (0.0523 + 0.0190i)7-s + (0.0307 − 0.174i)9-s + (0.725 − 1.25i)11-s + (0.532 − 0.0938i)13-s + (0.132 + 0.364i)15-s + (−0.833 − 0.147i)17-s + (0.994 + 1.18i)19-s + (−0.0567 + 0.0206i)21-s + (0.754 − 0.435i)23-s + (0.667 + 0.560i)25-s + (−0.446 − 0.773i)27-s + (0.488 + 0.282i)29-s + 1.46i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.874 - 0.484i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 592 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.874 - 0.484i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(592\)    =    \(2^{4} \cdot 37\)
Sign: $0.874 - 0.484i$
Analytic conductor: \(4.72714\)
Root analytic conductor: \(2.17419\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{592} (225, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 592,\ (\ :1/2),\ 0.874 - 0.484i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.16230 + 0.300491i\)
\(L(\frac12)\) \(\approx\) \(1.16230 + 0.300491i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
37 \( 1 + (-6.08 + 0.0772i)T \)
good3 \( 1 + (1.43 - 1.20i)T + (0.520 - 2.95i)T^{2} \)
5 \( 1 + (-0.273 + 0.751i)T + (-3.83 - 3.21i)T^{2} \)
7 \( 1 + (-0.138 - 0.0503i)T + (5.36 + 4.49i)T^{2} \)
11 \( 1 + (-2.40 + 4.16i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-1.91 + 0.338i)T + (12.2 - 4.44i)T^{2} \)
17 \( 1 + (3.43 + 0.606i)T + (15.9 + 5.81i)T^{2} \)
19 \( 1 + (-4.33 - 5.16i)T + (-3.29 + 18.7i)T^{2} \)
23 \( 1 + (-3.61 + 2.08i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.63 - 1.51i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 - 8.13iT - 31T^{2} \)
41 \( 1 + (-0.676 - 3.83i)T + (-38.5 + 14.0i)T^{2} \)
43 \( 1 + 8.10iT - 43T^{2} \)
47 \( 1 + (-4.16 - 7.22i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-10.2 + 3.72i)T + (40.6 - 34.0i)T^{2} \)
59 \( 1 + (-2.96 - 8.14i)T + (-45.1 + 37.9i)T^{2} \)
61 \( 1 + (0.346 - 0.0610i)T + (57.3 - 20.8i)T^{2} \)
67 \( 1 + (-2.25 - 0.820i)T + (51.3 + 43.0i)T^{2} \)
71 \( 1 + (5.34 - 4.48i)T + (12.3 - 69.9i)T^{2} \)
73 \( 1 - 1.13T + 73T^{2} \)
79 \( 1 + (0.646 - 1.77i)T + (-60.5 - 50.7i)T^{2} \)
83 \( 1 + (-1.71 + 9.75i)T + (-77.9 - 28.3i)T^{2} \)
89 \( 1 + (3.45 + 9.50i)T + (-68.1 + 57.2i)T^{2} \)
97 \( 1 + (-5.08 + 2.93i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.86474360781224473069257141914, −10.04780078039494815417410689613, −8.964649468682753048643288810202, −8.402728684156812874937734058197, −6.99296481989517404694741999438, −5.96033993630872996963487321569, −5.30347711926278923544833471218, −4.28033074838558931397271639849, −3.16063241565705696535816401869, −1.11515226289528156380316408021, 1.02715063817588468812040291575, 2.48988310008906369537264673117, 4.09429077917652288385387567056, 5.15437880444825042462437756886, 6.36380625722164643455527081123, 6.81607354713785811301243592459, 7.65153011278959713786716962419, 9.045114051077443448122716688744, 9.666496918909851973571995198661, 10.91892932392652462271623236248

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