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Properties

Label 2-592-148.103-c1-0-17
Degree $2$
Conductor $592$
Sign $-0.682 + 0.731i$
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  + (1.55 − 2.69i)3-s + (−0.386 − 1.44i)5-s + (1.53 + 0.888i)7-s + (−3.34 − 5.80i)9-s − 4.50·11-s + (−0.816 − 3.04i)13-s + (−4.48 − 1.20i)15-s + (1.00 + 0.269i)17-s + (−2.33 + 0.624i)19-s + (4.79 − 2.76i)21-s + (5.24 + 5.24i)23-s + (2.40 − 1.38i)25-s − 11.5·27-s + (3.77 + 3.77i)29-s + (4.22 − 4.22i)31-s + ⋯
L(s)  = 1  + (0.898 − 1.55i)3-s + (−0.172 − 0.644i)5-s + (0.581 + 0.335i)7-s + (−1.11 − 1.93i)9-s − 1.35·11-s + (−0.226 − 0.845i)13-s + (−1.15 − 0.310i)15-s + (0.243 + 0.0652i)17-s + (−0.534 + 0.143i)19-s + (1.04 − 0.603i)21-s + (1.09 + 1.09i)23-s + (0.480 − 0.277i)25-s − 2.21·27-s + (0.701 + 0.701i)29-s + (0.758 − 0.758i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.682 + 0.731i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 592 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.682 + 0.731i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(592\)    =    \(2^{4} \cdot 37\)
Sign: $-0.682 + 0.731i$
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: \((2,\ 592,\ (\ :1/2),\ -0.682 + 0.731i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.703501 - 1.61824i\)
\(L(\frac12)\) \(\approx\) \(0.703501 - 1.61824i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.55 + 5.87i)T \)
good3 \( 1 + (-1.55 + 2.69i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (0.386 + 1.44i)T + (-4.33 + 2.5i)T^{2} \)
7 \( 1 + (-1.53 - 0.888i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + 4.50T + 11T^{2} \)
13 \( 1 + (0.816 + 3.04i)T + (-11.2 + 6.5i)T^{2} \)
17 \( 1 + (-1.00 - 0.269i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (2.33 - 0.624i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + (-5.24 - 5.24i)T + 23iT^{2} \)
29 \( 1 + (-3.77 - 3.77i)T + 29iT^{2} \)
31 \( 1 + (-4.22 + 4.22i)T - 31iT^{2} \)
41 \( 1 + (-7.21 - 4.16i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (7.73 + 7.73i)T + 43iT^{2} \)
47 \( 1 - 5.13iT - 47T^{2} \)
53 \( 1 + (3.45 + 5.98i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (2.71 - 10.1i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (-4.78 + 1.28i)T + (52.8 - 30.5i)T^{2} \)
67 \( 1 + (6.03 - 10.4i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-5.09 - 2.93i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + 1.33iT - 73T^{2} \)
79 \( 1 + (-6.53 + 1.75i)T + (68.4 - 39.5i)T^{2} \)
83 \( 1 + (-4.89 + 2.82i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (-1.46 + 5.45i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (1.23 - 1.23i)T - 97iT^{2} \)
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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

−10.35549457082083657763022133001, −9.092399739952990753819917735803, −8.308854075435017403218921988926, −7.87952845452283322661748582845, −7.09653886315452236539656416511, −5.82886146637715597732391278680, −4.91034867918977228799065742756, −3.15253282301555073394183105033, −2.25479396380619858267823628958, −0.913069848151797284625122282143, 2.49845279458301418687642736368, 3.23793244204170804600263628237, 4.61341582785696808180931597335, 4.87842486087682476823641035197, 6.57465760046609046236947142090, 7.79631802932735903901076524365, 8.419399098338388798030822971904, 9.341361589860581874176964143550, 10.26841087108397712044757648821, 10.71276794888891728173653435726

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