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Properties

Label 2-592-148.103-c1-0-14
Degree $2$
Conductor $592$
Sign $-0.171 + 0.985i$
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  + (1.28 − 2.22i)3-s + (−0.133 − 0.5i)5-s + (0.256 + 0.148i)7-s + (−1.80 − 3.13i)9-s + 4.15·11-s + (−0.973 − 3.63i)13-s + (−1.28 − 0.344i)15-s + (−3.07 − 0.823i)17-s + (0.880 − 0.236i)19-s + (0.660 − 0.381i)21-s + (−1.05 − 1.05i)23-s + (4.09 − 2.36i)25-s − 1.58·27-s + (−2.32 − 2.32i)29-s + (−4.09 + 4.09i)31-s + ⋯
L(s)  = 1  + (0.742 − 1.28i)3-s + (−0.0599 − 0.223i)5-s + (0.0970 + 0.0560i)7-s + (−0.602 − 1.04i)9-s + 1.25·11-s + (−0.269 − 1.00i)13-s + (−0.332 − 0.0889i)15-s + (−0.745 − 0.199i)17-s + (0.202 − 0.0541i)19-s + (0.144 − 0.0832i)21-s + (−0.218 − 0.218i)23-s + (0.819 − 0.473i)25-s − 0.305·27-s + (−0.431 − 0.431i)29-s + (−0.735 + 0.735i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.171 + 0.985i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 592 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.171 + 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(592\)    =    \(2^{4} \cdot 37\)
Sign: $-0.171 + 0.985i$
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: \((2,\ 592,\ (\ :1/2),\ -0.171 + 0.985i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.20487 - 1.43277i\)
\(L(\frac12)\) \(\approx\) \(1.20487 - 1.43277i\)
\(L(\frac{3}{2})\) 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 \( 1 + (4.04 - 4.54i)T \)
good3 \( 1 + (-1.28 + 2.22i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (0.133 + 0.5i)T + (-4.33 + 2.5i)T^{2} \)
7 \( 1 + (-0.256 - 0.148i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 - 4.15T + 11T^{2} \)
13 \( 1 + (0.973 + 3.63i)T + (-11.2 + 6.5i)T^{2} \)
17 \( 1 + (3.07 + 0.823i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (-0.880 + 0.236i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + (1.05 + 1.05i)T + 23iT^{2} \)
29 \( 1 + (2.32 + 2.32i)T + 29iT^{2} \)
31 \( 1 + (4.09 - 4.09i)T - 31iT^{2} \)
41 \( 1 + (2.06 + 1.18i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-5.44 - 5.44i)T + 43iT^{2} \)
47 \( 1 - 5.23iT - 47T^{2} \)
53 \( 1 + (-3.22 - 5.58i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (0.837 - 3.12i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (-3.80 + 1.01i)T + (52.8 - 30.5i)T^{2} \)
67 \( 1 + (-0.0605 + 0.104i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-1.18 - 0.684i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 - 3.46iT - 73T^{2} \)
79 \( 1 + (-10.4 + 2.79i)T + (68.4 - 39.5i)T^{2} \)
83 \( 1 + (-10.4 + 6.02i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (2.86 - 10.6i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (-9.25 + 9.25i)T - 97iT^{2} \)
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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

−10.44496811897900672699388638007, −9.228695042125239089387228500778, −8.617790231822400289515853110729, −7.77642756179243606599940285310, −6.95693567415072735888787655460, −6.19717746399122073227778323372, −4.83218082618924326512536405255, −3.45106220796283765739766076670, −2.29653234710185842530006201522, −1.05361435434984545407742730694, 2.03185114906275654662792879732, 3.53088751110522306535809233875, 4.09850043369053134259836155389, 5.10774648428150062841697444608, 6.49236988639889893696347518046, 7.38720925681324974525211520186, 8.788430384602383593519897288100, 9.104177809592465061288439785839, 9.876947039416675214503836127066, 10.87280136069202555882463920930

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