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Properties

Label 2-592-37.26-c1-0-11
Degree $2$
Conductor $592$
Sign $0.0800 + 0.996i$
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  + (−1.52 + 2.64i)3-s + (−0.629 + 1.09i)5-s + (1.52 − 2.64i)7-s + (−3.15 − 5.46i)9-s − 5.31·11-s + (−1 + 1.73i)13-s + (−1.92 − 3.32i)15-s + (−2.55 − 4.41i)17-s + (2.39 − 4.14i)19-s + (4.65 + 8.06i)21-s − 1.74·23-s + (1.70 + 2.95i)25-s + 10.1·27-s − 4.05·29-s + 0.791·31-s + ⋯
L(s)  = 1  + (−0.880 + 1.52i)3-s + (−0.281 + 0.487i)5-s + (0.576 − 0.998i)7-s + (−1.05 − 1.82i)9-s − 1.60·11-s + (−0.277 + 0.480i)13-s + (−0.496 − 0.859i)15-s + (−0.618 − 1.07i)17-s + (0.549 − 0.952i)19-s + (1.01 + 1.75i)21-s − 0.362·23-s + (0.341 + 0.591i)25-s + 1.94·27-s − 0.752·29-s + 0.142·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0800 + 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 592 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0800 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(592\)    =    \(2^{4} \cdot 37\)
Sign: $0.0800 + 0.996i$
Analytic conductor: \(4.72714\)
Root analytic conductor: \(2.17419\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{592} (433, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 592,\ (\ :1/2),\ 0.0800 + 0.996i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.168922 - 0.155904i\)
\(L(\frac12)\) \(\approx\) \(0.168922 - 0.155904i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.37 + 5.92i)T \)
good3 \( 1 + (1.52 - 2.64i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (0.629 - 1.09i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (-1.52 + 2.64i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + 5.31T + 11T^{2} \)
13 \( 1 + (1 - 1.73i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (2.55 + 4.41i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.39 + 4.14i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + 1.74T + 23T^{2} \)
29 \( 1 + 4.05T + 29T^{2} \)
31 \( 1 - 0.791T + 31T^{2} \)
41 \( 1 + (-0.104 + 0.180i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 - 4.36T + 43T^{2} \)
47 \( 1 - 3.20T + 47T^{2} \)
53 \( 1 + (5.65 + 9.79i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (6.36 + 11.0i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.02 - 5.24i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (4.65 - 8.06i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (3.52 - 6.10i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + 3.31T + 73T^{2} \)
79 \( 1 + (5.70 - 9.88i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (1.78 + 3.09i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (1.63 + 2.83i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 2.20T + 97T^{2} \)
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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

−10.66605997322097035939441343389, −9.839567530386850695037426548908, −9.049019789148025842559211643622, −7.64508904210815898749832072857, −6.95496918061399041116761996157, −5.49249623491808098145879321593, −4.85607790151428117473268584417, −4.06825599376472221033570636737, −2.84350739528966741996032197202, −0.13988418377948126236933168358, 1.58710763829799540582402163984, 2.64697035514629600978281620335, 4.75616101250111957534249869366, 5.62119652833194544403168118627, 6.14943504237725337297848616891, 7.61024298159654779008058507019, 7.936086900749636173808673450868, 8.762426996433994841871947701281, 10.36078246180686749565139552665, 11.03207155624660883812861879113

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