Properties

Label 2-592-37.34-c1-0-17
Degree $2$
Conductor $592$
Sign $-0.765 + 0.642i$
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  + (2.41 − 2.02i)3-s + (−1.43 − 0.524i)5-s + (−4.53 − 1.65i)7-s + (1.20 − 6.81i)9-s + (−0.546 + 0.947i)11-s + (0.307 + 1.74i)13-s + (−4.53 + 1.65i)15-s + (0.511 − 2.90i)17-s + (0.632 − 0.530i)19-s + (−14.2 + 5.19i)21-s + (−0.121 − 0.210i)23-s + (−2.03 − 1.70i)25-s + (−6.17 − 10.6i)27-s + (2.78 − 4.82i)29-s + 5.73·31-s + ⋯
L(s)  = 1  + (1.39 − 1.16i)3-s + (−0.643 − 0.234i)5-s + (−1.71 − 0.623i)7-s + (0.400 − 2.27i)9-s + (−0.164 + 0.285i)11-s + (0.0852 + 0.483i)13-s + (−1.17 + 0.426i)15-s + (0.124 − 0.703i)17-s + (0.145 − 0.121i)19-s + (−3.11 + 1.13i)21-s + (−0.0253 − 0.0439i)23-s + (−0.406 − 0.341i)25-s + (−1.18 − 2.05i)27-s + (0.517 − 0.896i)29-s + 1.03·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.765 + 0.642i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 592 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.765 + 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(592\)    =    \(2^{4} \cdot 37\)
Sign: $-0.765 + 0.642i$
Analytic conductor: \(4.72714\)
Root analytic conductor: \(2.17419\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{592} (145, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 592,\ (\ :1/2),\ -0.765 + 0.642i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.509075 - 1.39821i\)
\(L(\frac12)\) \(\approx\) \(0.509075 - 1.39821i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-5.84 + 1.69i)T \)
good3 \( 1 + (-2.41 + 2.02i)T + (0.520 - 2.95i)T^{2} \)
5 \( 1 + (1.43 + 0.524i)T + (3.83 + 3.21i)T^{2} \)
7 \( 1 + (4.53 + 1.65i)T + (5.36 + 4.49i)T^{2} \)
11 \( 1 + (0.546 - 0.947i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.307 - 1.74i)T + (-12.2 + 4.44i)T^{2} \)
17 \( 1 + (-0.511 + 2.90i)T + (-15.9 - 5.81i)T^{2} \)
19 \( 1 + (-0.632 + 0.530i)T + (3.29 - 18.7i)T^{2} \)
23 \( 1 + (0.121 + 0.210i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2.78 + 4.82i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 5.73T + 31T^{2} \)
41 \( 1 + (-0.505 - 2.86i)T + (-38.5 + 14.0i)T^{2} \)
43 \( 1 + 4.37T + 43T^{2} \)
47 \( 1 + (-1.13 - 1.96i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (5.77 - 2.10i)T + (40.6 - 34.0i)T^{2} \)
59 \( 1 + (-8.29 + 3.01i)T + (45.1 - 37.9i)T^{2} \)
61 \( 1 + (2.04 + 11.5i)T + (-57.3 + 20.8i)T^{2} \)
67 \( 1 + (-8.44 - 3.07i)T + (51.3 + 43.0i)T^{2} \)
71 \( 1 + (-0.933 + 0.783i)T + (12.3 - 69.9i)T^{2} \)
73 \( 1 - 8.79T + 73T^{2} \)
79 \( 1 + (-2.02 - 0.736i)T + (60.5 + 50.7i)T^{2} \)
83 \( 1 + (0.884 - 5.01i)T + (-77.9 - 28.3i)T^{2} \)
89 \( 1 + (8.43 - 3.07i)T + (68.1 - 57.2i)T^{2} \)
97 \( 1 + (2.17 + 3.77i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.803787441511387810960439782633, −9.557845287286114476799454544566, −8.387258210269194051008285184468, −7.71623249108610928140694158977, −6.87659409845920896916420491609, −6.32152395263150137253606645787, −4.27204546272412706438423495864, −3.34626253037780111006728516714, −2.47493664607111040215590662903, −0.69396677884421068136558539205, 2.65241078998166089296148267718, 3.32466972213594792248490735753, 3.99871579930797199023871916573, 5.38708944008763645718428501524, 6.57475308585445764829858271036, 7.82269062083699681751483224817, 8.533105164738650482112614818702, 9.309854823849820780888774408971, 10.02397097130445128305294021439, 10.59872040992885420252026876692

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