Properties

Label 2-592-37.33-c1-0-17
Degree $2$
Conductor $592$
Sign $-0.685 - 0.727i$
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  + (−0.504 − 2.86i)3-s + (0.266 − 0.223i)5-s + (−0.773 + 0.649i)7-s + (−5.12 + 1.86i)9-s + (−2.73 + 4.73i)11-s + (−5.97 − 2.17i)13-s + (−0.773 − 0.649i)15-s + (−0.490 + 0.178i)17-s + (−0.869 − 4.93i)19-s + (2.24 + 1.88i)21-s + (0.721 + 1.24i)23-s + (−0.847 + 4.80i)25-s + (3.56 + 6.18i)27-s + (3.16 − 5.48i)29-s − 3.30·31-s + ⋯
L(s)  = 1  + (−0.291 − 1.65i)3-s + (0.118 − 0.0998i)5-s + (−0.292 + 0.245i)7-s + (−1.70 + 0.621i)9-s + (−0.823 + 1.42i)11-s + (−1.65 − 0.603i)13-s + (−0.199 − 0.167i)15-s + (−0.119 + 0.0433i)17-s + (−0.199 − 1.13i)19-s + (0.490 + 0.411i)21-s + (0.150 + 0.260i)23-s + (−0.169 + 0.961i)25-s + (0.686 + 1.18i)27-s + (0.588 − 1.01i)29-s − 0.594·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.685 - 0.727i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 592 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.685 - 0.727i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(592\)    =    \(2^{4} \cdot 37\)
Sign: $-0.685 - 0.727i$
Analytic conductor: \(4.72714\)
Root analytic conductor: \(2.17419\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{592} (33, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 592,\ (\ :1/2),\ -0.685 - 0.727i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.100672 + 0.233262i\)
\(L(\frac12)\) \(\approx\) \(0.100672 + 0.233262i\)
\(L(\frac{3}{2})\) 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 \( 1 + (5.88 + 1.53i)T \)
good3 \( 1 + (0.504 + 2.86i)T + (-2.81 + 1.02i)T^{2} \)
5 \( 1 + (-0.266 + 0.223i)T + (0.868 - 4.92i)T^{2} \)
7 \( 1 + (0.773 - 0.649i)T + (1.21 - 6.89i)T^{2} \)
11 \( 1 + (2.73 - 4.73i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (5.97 + 2.17i)T + (9.95 + 8.35i)T^{2} \)
17 \( 1 + (0.490 - 0.178i)T + (13.0 - 10.9i)T^{2} \)
19 \( 1 + (0.869 + 4.93i)T + (-17.8 + 6.49i)T^{2} \)
23 \( 1 + (-0.721 - 1.24i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3.16 + 5.48i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 3.30T + 31T^{2} \)
41 \( 1 + (-1.03 - 0.376i)T + (31.4 + 26.3i)T^{2} \)
43 \( 1 - 6.27T + 43T^{2} \)
47 \( 1 + (5.71 + 9.90i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-0.0706 - 0.0593i)T + (9.20 + 52.1i)T^{2} \)
59 \( 1 + (2.29 + 1.92i)T + (10.2 + 58.1i)T^{2} \)
61 \( 1 + (3.58 + 1.30i)T + (46.7 + 39.2i)T^{2} \)
67 \( 1 + (-0.892 + 0.748i)T + (11.6 - 65.9i)T^{2} \)
71 \( 1 + (0.0543 + 0.308i)T + (-66.7 + 24.2i)T^{2} \)
73 \( 1 - 13.2T + 73T^{2} \)
79 \( 1 + (7.12 - 5.98i)T + (13.7 - 77.7i)T^{2} \)
83 \( 1 + (-8.89 + 3.23i)T + (63.5 - 53.3i)T^{2} \)
89 \( 1 + (12.0 + 10.1i)T + (15.4 + 87.6i)T^{2} \)
97 \( 1 + (-0.464 - 0.804i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.09549296878741460446785199736, −9.273582278928606413377580178146, −7.985866727234218285734470472479, −7.32362823428473977659558828376, −6.83205503293447364818405895494, −5.57802976796833841614830508128, −4.82315444603388821632649442557, −2.71569965502136160346290821214, −1.98170287410279850838554764756, −0.13435432368220784642501203380, 2.72633033650310616883607574769, 3.72226980052587728401364647613, 4.76442106317189556240436001591, 5.49803942277326384576962003509, 6.52134738601422476978381382059, 7.88517755490953668580029667568, 8.865054321544380146076633821044, 9.681847500996805418682414847179, 10.41192257986867220250812849281, 10.84137802608747936579804243084

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