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Properties

Label 2-592-37.12-c1-0-8
Degree $2$
Conductor $592$
Sign $0.492 - 0.870i$
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  + (2.22 + 1.86i)3-s + (−0.243 + 0.0887i)5-s + (2.73 − 0.995i)7-s + (0.942 + 5.34i)9-s + (0.152 + 0.263i)11-s + (−0.261 + 1.48i)13-s + (−0.707 − 0.257i)15-s + (0.433 + 2.45i)17-s + (−3.74 − 3.14i)19-s + (7.93 + 2.88i)21-s + (2.76 − 4.79i)23-s + (−3.77 + 3.17i)25-s + (−3.52 + 6.10i)27-s + (−2.12 − 3.68i)29-s + 7.67·31-s + ⋯
L(s)  = 1  + (1.28 + 1.07i)3-s + (−0.109 + 0.0396i)5-s + (1.03 − 0.376i)7-s + (0.314 + 1.78i)9-s + (0.0458 + 0.0794i)11-s + (−0.0724 + 0.410i)13-s + (−0.182 − 0.0665i)15-s + (0.105 + 0.595i)17-s + (−0.859 − 0.721i)19-s + (1.73 + 0.630i)21-s + (0.577 − 1.00i)23-s + (−0.755 + 0.634i)25-s + (−0.678 + 1.17i)27-s + (−0.394 − 0.683i)29-s + 1.37·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.492 - 0.870i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 592 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.492 - 0.870i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(592\)    =    \(2^{4} \cdot 37\)
Sign: $0.492 - 0.870i$
Analytic conductor: \(4.72714\)
Root analytic conductor: \(2.17419\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{592} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 592,\ (\ :1/2),\ 0.492 - 0.870i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.04267 + 1.19052i\)
\(L(\frac12)\) \(\approx\) \(2.04267 + 1.19052i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.609 + 6.05i)T \)
good3 \( 1 + (-2.22 - 1.86i)T + (0.520 + 2.95i)T^{2} \)
5 \( 1 + (0.243 - 0.0887i)T + (3.83 - 3.21i)T^{2} \)
7 \( 1 + (-2.73 + 0.995i)T + (5.36 - 4.49i)T^{2} \)
11 \( 1 + (-0.152 - 0.263i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (0.261 - 1.48i)T + (-12.2 - 4.44i)T^{2} \)
17 \( 1 + (-0.433 - 2.45i)T + (-15.9 + 5.81i)T^{2} \)
19 \( 1 + (3.74 + 3.14i)T + (3.29 + 18.7i)T^{2} \)
23 \( 1 + (-2.76 + 4.79i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (2.12 + 3.68i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 7.67T + 31T^{2} \)
41 \( 1 + (1.39 - 7.92i)T + (-38.5 - 14.0i)T^{2} \)
43 \( 1 - 4.49T + 43T^{2} \)
47 \( 1 + (5.38 - 9.33i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (9.86 + 3.59i)T + (40.6 + 34.0i)T^{2} \)
59 \( 1 + (12.7 + 4.62i)T + (45.1 + 37.9i)T^{2} \)
61 \( 1 + (-0.615 + 3.49i)T + (-57.3 - 20.8i)T^{2} \)
67 \( 1 + (-9.78 + 3.55i)T + (51.3 - 43.0i)T^{2} \)
71 \( 1 + (3.12 + 2.62i)T + (12.3 + 69.9i)T^{2} \)
73 \( 1 + 14.5T + 73T^{2} \)
79 \( 1 + (-11.5 + 4.19i)T + (60.5 - 50.7i)T^{2} \)
83 \( 1 + (-0.411 - 2.33i)T + (-77.9 + 28.3i)T^{2} \)
89 \( 1 + (-14.0 - 5.09i)T + (68.1 + 57.2i)T^{2} \)
97 \( 1 + (-5.21 + 9.03i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.82007321703535166433878438573, −9.796544747560683307924490823744, −9.123040916745941174797773737916, −8.207199473854407366627018636231, −7.74486024028098112801172134395, −6.34497954166927947192751546373, −4.66313308741256526229962837946, −4.39065721841568856267179315725, −3.13618198985180915060313394646, −1.94632662816819299222518592056, 1.41524552270191682576594413103, 2.42559443857155544836017399038, 3.54077441102910522927992962447, 4.91823839297659238339390534718, 6.17606319415572848761482405173, 7.28115075822311739466055302245, 7.990718413185672179035436406439, 8.506243296559766110017114984526, 9.363906826390107067180417082979, 10.49152537151519142648895685851

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