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Properties

Label 2-592-148.119-c1-0-12
Degree $2$
Conductor $592$
Sign $0.702 + 0.711i$
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  + (1.13 − 1.96i)3-s + (−1.86 + 0.5i)5-s + (3.74 + 2.16i)7-s + (−1.08 − 1.87i)9-s + 0.384·11-s + (5.14 − 1.37i)13-s + (−1.13 + 4.24i)15-s + (0.927 − 3.46i)17-s + (−0.445 − 1.66i)19-s + (8.51 − 4.91i)21-s + (−2.79 + 2.79i)23-s + (−1.09 + 0.633i)25-s + 1.88·27-s + (1.96 − 1.96i)29-s + (−1.74 − 1.74i)31-s + ⋯
L(s)  = 1  + (0.656 − 1.13i)3-s + (−0.834 + 0.223i)5-s + (1.41 + 0.816i)7-s + (−0.361 − 0.626i)9-s + 0.115·11-s + (1.42 − 0.382i)13-s + (−0.293 + 1.09i)15-s + (0.225 − 0.839i)17-s + (−0.102 − 0.381i)19-s + (1.85 − 1.07i)21-s + (−0.583 + 0.583i)23-s + (−0.219 + 0.126i)25-s + 0.363·27-s + (0.364 − 0.364i)29-s + (−0.314 − 0.314i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.702 + 0.711i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 592 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.702 + 0.711i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(592\)    =    \(2^{4} \cdot 37\)
Sign: $0.702 + 0.711i$
Analytic conductor: \(4.72714\)
Root analytic conductor: \(2.17419\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{592} (415, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 592,\ (\ :1/2),\ 0.702 + 0.711i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.76991 - 0.739919i\)
\(L(\frac12)\) \(\approx\) \(1.76991 - 0.739919i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-6.07 + 0.352i)T \)
good3 \( 1 + (-1.13 + 1.96i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (1.86 - 0.5i)T + (4.33 - 2.5i)T^{2} \)
7 \( 1 + (-3.74 - 2.16i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 - 0.384T + 11T^{2} \)
13 \( 1 + (-5.14 + 1.37i)T + (11.2 - 6.5i)T^{2} \)
17 \( 1 + (-0.927 + 3.46i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (0.445 + 1.66i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (2.79 - 2.79i)T - 23iT^{2} \)
29 \( 1 + (-1.96 + 1.96i)T - 29iT^{2} \)
31 \( 1 + (1.74 + 1.74i)T + 31iT^{2} \)
41 \( 1 + (8.36 + 4.82i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.224 + 0.224i)T - 43iT^{2} \)
47 \( 1 - 8.99iT - 47T^{2} \)
53 \( 1 + (2.60 + 4.51i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (6.32 + 1.69i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (3.65 + 13.6i)T + (-52.8 + 30.5i)T^{2} \)
67 \( 1 + (2.66 - 4.60i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-11.8 - 6.84i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 - 3.46iT - 73T^{2} \)
79 \( 1 + (-4.12 - 15.3i)T + (-68.4 + 39.5i)T^{2} \)
83 \( 1 + (7.10 - 4.10i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (7.62 + 2.04i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (-5.73 - 5.73i)T + 97iT^{2} \)
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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

−11.00286765055209337288934322858, −9.419151789094582000184781380208, −8.320199844053295024801323675473, −8.095250143712146903562337525522, −7.29925527171371601926704943755, −6.16630415225797656808986844719, −5.03535851740511786390029298924, −3.70616633947543611599592330713, −2.45892569112504143168286469815, −1.31712557613753495588159198610, 1.49994401506136851037358412587, 3.49551971207815151484676155243, 4.14483816728361599286381148559, 4.73471332233961211702783181395, 6.21314437366945054272364317087, 7.61618118142015064501689918157, 8.373663131462109251052744422799, 8.756209546419441196763841996385, 10.10199006107341594223626115372, 10.70071903358893265209595573487

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