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Properties

Label 2-592-37.9-c1-0-16
Degree $2$
Conductor $592$
Sign $-0.942 + 0.335i$
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  + (0.453 − 2.56i)3-s + (0.356 + 0.299i)5-s + (−1.67 − 1.40i)7-s + (−3.57 − 1.30i)9-s + (−1.28 − 2.23i)11-s + (−1.81 + 0.659i)13-s + (0.930 − 0.780i)15-s + (−3.45 − 1.25i)17-s + (−0.195 + 1.11i)19-s + (−4.37 + 3.67i)21-s + (1.76 − 3.05i)23-s + (−0.830 − 4.71i)25-s + (−1.05 + 1.82i)27-s + (1.87 + 3.25i)29-s + 0.372·31-s + ⋯
L(s)  = 1  + (0.261 − 1.48i)3-s + (0.159 + 0.133i)5-s + (−0.634 − 0.532i)7-s + (−1.19 − 0.433i)9-s + (−0.388 − 0.673i)11-s + (−0.502 + 0.182i)13-s + (0.240 − 0.201i)15-s + (−0.837 − 0.304i)17-s + (−0.0449 + 0.254i)19-s + (−0.955 + 0.801i)21-s + (0.367 − 0.636i)23-s + (−0.166 − 0.942i)25-s + (−0.202 + 0.350i)27-s + (0.348 + 0.603i)29-s + 0.0668·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.942 + 0.335i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 592 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.942 + 0.335i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(592\)    =    \(2^{4} \cdot 37\)
Sign: $-0.942 + 0.335i$
Analytic conductor: \(4.72714\)
Root analytic conductor: \(2.17419\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{592} (305, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 592,\ (\ :1/2),\ -0.942 + 0.335i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.191070 - 1.10634i\)
\(L(\frac12)\) \(\approx\) \(0.191070 - 1.10634i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-5.93 - 1.31i)T \)
good3 \( 1 + (-0.453 + 2.56i)T + (-2.81 - 1.02i)T^{2} \)
5 \( 1 + (-0.356 - 0.299i)T + (0.868 + 4.92i)T^{2} \)
7 \( 1 + (1.67 + 1.40i)T + (1.21 + 6.89i)T^{2} \)
11 \( 1 + (1.28 + 2.23i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (1.81 - 0.659i)T + (9.95 - 8.35i)T^{2} \)
17 \( 1 + (3.45 + 1.25i)T + (13.0 + 10.9i)T^{2} \)
19 \( 1 + (0.195 - 1.11i)T + (-17.8 - 6.49i)T^{2} \)
23 \( 1 + (-1.76 + 3.05i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.87 - 3.25i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 0.372T + 31T^{2} \)
41 \( 1 + (-7.81 + 2.84i)T + (31.4 - 26.3i)T^{2} \)
43 \( 1 + 9.43T + 43T^{2} \)
47 \( 1 + (2.11 - 3.67i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (7.09 - 5.95i)T + (9.20 - 52.1i)T^{2} \)
59 \( 1 + (3.98 - 3.34i)T + (10.2 - 58.1i)T^{2} \)
61 \( 1 + (-12.3 + 4.50i)T + (46.7 - 39.2i)T^{2} \)
67 \( 1 + (10.4 + 8.73i)T + (11.6 + 65.9i)T^{2} \)
71 \( 1 + (-2.05 + 11.6i)T + (-66.7 - 24.2i)T^{2} \)
73 \( 1 - 1.32T + 73T^{2} \)
79 \( 1 + (-13.0 - 10.9i)T + (13.7 + 77.7i)T^{2} \)
83 \( 1 + (-12.4 - 4.54i)T + (63.5 + 53.3i)T^{2} \)
89 \( 1 + (-11.7 + 9.84i)T + (15.4 - 87.6i)T^{2} \)
97 \( 1 + (-4.27 + 7.40i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.36288176796275893039204828904, −9.302126719462339530625796985692, −8.312750407823377605720067743479, −7.57305430131222601742646897993, −6.64762759451849475899699936039, −6.20616693372570800749714221475, −4.68857425915164171179141156285, −3.14585684154843485433817641387, −2.17528283964442668346607311065, −0.58137469277643524338074613814, 2.40405906049935035495795566968, 3.47600432272654766371711895347, 4.57614427621028716051537755123, 5.28875971563942790479892310802, 6.42251710698270209764854046978, 7.67141534915541385001076352226, 8.802416533014729090542578825033, 9.524286986183979162906173265150, 9.930380207428329722109899214322, 10.88487495025554616299943752696

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