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Properties

Label 2-592-148.103-c1-0-18
Degree $2$
Conductor $592$
Sign $-0.914 + 0.405i$
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  + (1.36 − 2.36i)3-s + (−1.06 − 3.98i)5-s + (−2.33 − 1.35i)7-s + (−2.23 − 3.86i)9-s + 3.86·11-s + (1.36 + 5.09i)13-s + (−10.8 − 2.91i)15-s + (6.70 + 1.79i)17-s + (−3.48 + 0.934i)19-s + (−6.38 + 3.68i)21-s + (−3.40 − 3.40i)23-s + (−10.4 + 6.01i)25-s − 4.00·27-s + (−0.545 − 0.545i)29-s + (4.98 − 4.98i)31-s + ⋯
L(s)  = 1  + (0.788 − 1.36i)3-s + (−0.477 − 1.78i)5-s + (−0.883 − 0.510i)7-s + (−0.744 − 1.28i)9-s + 1.16·11-s + (0.378 + 1.41i)13-s + (−2.81 − 0.753i)15-s + (1.62 + 0.435i)17-s + (−0.799 + 0.214i)19-s + (−1.39 + 0.804i)21-s + (−0.709 − 0.709i)23-s + (−2.08 + 1.20i)25-s − 0.769·27-s + (−0.101 − 0.101i)29-s + (0.895 − 0.895i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.914 + 0.405i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 592 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.914 + 0.405i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(592\)    =    \(2^{4} \cdot 37\)
Sign: $-0.914 + 0.405i$
Analytic conductor: \(4.72714\)
Root analytic conductor: \(2.17419\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{592} (399, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 592,\ (\ :1/2),\ -0.914 + 0.405i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.334619 - 1.57985i\)
\(L(\frac12)\) \(\approx\) \(0.334619 - 1.57985i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.28 - 5.63i)T \)
good3 \( 1 + (-1.36 + 2.36i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (1.06 + 3.98i)T + (-4.33 + 2.5i)T^{2} \)
7 \( 1 + (2.33 + 1.35i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 - 3.86T + 11T^{2} \)
13 \( 1 + (-1.36 - 5.09i)T + (-11.2 + 6.5i)T^{2} \)
17 \( 1 + (-6.70 - 1.79i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (3.48 - 0.934i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + (3.40 + 3.40i)T + 23iT^{2} \)
29 \( 1 + (0.545 + 0.545i)T + 29iT^{2} \)
31 \( 1 + (-4.98 + 4.98i)T - 31iT^{2} \)
41 \( 1 + (-1.5 - 0.866i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (3.58 + 3.58i)T + 43iT^{2} \)
47 \( 1 + 4.03iT - 47T^{2} \)
53 \( 1 + (-0.700 - 1.21i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (2.28 - 8.52i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (-5.30 + 1.42i)T + (52.8 - 30.5i)T^{2} \)
67 \( 1 + (-4.93 + 8.54i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (4.75 + 2.74i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + 4.34iT - 73T^{2} \)
79 \( 1 + (3.71 - 0.995i)T + (68.4 - 39.5i)T^{2} \)
83 \( 1 + (-0.945 + 0.545i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (1.29 - 4.81i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (-1.14 + 1.14i)T - 97iT^{2} \)
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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

−9.949705602510370557882493787602, −9.149302230061740009515646989567, −8.447862969963275897219589316393, −7.85436963309903382416062636970, −6.74053498411784187083737428661, −6.06540420955596330734973158796, −4.36325686151943476006996659280, −3.64775854047143110673456105067, −1.79814465320024482638956015440, −0.888211907242875537091394704783, 2.83761510737649381778577339883, 3.29439087377528411874698255226, 4.00161135306705562528864248081, 5.64727828507589701305866893672, 6.52647965843831308725707703719, 7.63918831412452356986174917632, 8.515769361093828199334492514588, 9.688129004461559716524961618501, 10.00964086707398161721798511465, 10.79750809373385901140066311739

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