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Properties

Label 2-592-37.26-c1-0-6
Degree $2$
Conductor $592$
Sign $0.508 - 0.861i$
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  + (0.403 − 0.698i)3-s + (0.0969 − 0.167i)5-s + (−2.07 + 3.59i)7-s + (1.17 + 2.03i)9-s + (1 − 1.73i)13-s + (−0.0781 − 0.135i)15-s + (2.65 + 4.60i)17-s + (−2.15 + 3.73i)19-s + (1.67 + 2.90i)21-s − 5.76·23-s + (2.48 + 4.29i)25-s + 4.31·27-s − 2.76·29-s + 5.61·31-s + (0.403 + 0.698i)35-s + ⋯
L(s)  = 1  + (0.232 − 0.403i)3-s + (0.0433 − 0.0751i)5-s + (−0.785 + 1.36i)7-s + (0.391 + 0.678i)9-s + (0.277 − 0.480i)13-s + (−0.0201 − 0.0349i)15-s + (0.644 + 1.11i)17-s + (−0.494 + 0.856i)19-s + (0.365 + 0.633i)21-s − 1.20·23-s + (0.496 + 0.859i)25-s + 0.829·27-s − 0.514·29-s + 1.00·31-s + (0.0681 + 0.117i)35-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.508 - 0.861i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 592 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.508 - 0.861i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(592\)    =    \(2^{4} \cdot 37\)
Sign: $0.508 - 0.861i$
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: \((2,\ 592,\ (\ :1/2),\ 0.508 - 0.861i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.20998 + 0.690823i\)
\(L(\frac12)\) \(\approx\) \(1.20998 + 0.690823i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.29 + 5.63i)T \)
good3 \( 1 + (-0.403 + 0.698i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (-0.0969 + 0.167i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (2.07 - 3.59i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 + (-1 + 1.73i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-2.65 - 4.60i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.15 - 3.73i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + 5.76T + 23T^{2} \)
29 \( 1 + 2.76T + 29T^{2} \)
31 \( 1 - 5.61T + 31T^{2} \)
41 \( 1 + (-0.981 + 1.69i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 - 9.89T + 43T^{2} \)
47 \( 1 - 1.61T + 47T^{2} \)
53 \( 1 + (0.675 + 1.16i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-2.15 - 3.73i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (6.38 - 11.0i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.387 + 0.671i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-5.04 + 8.73i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + 12.0T + 73T^{2} \)
79 \( 1 + (5.50 - 9.53i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (4.14 + 7.17i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (7.74 + 13.4i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 7.18T + 97T^{2} \)
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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

−10.66405249460506206311898458574, −9.993668912577922139570309437411, −8.957617219796295364367002229227, −8.225205351263809619566951047094, −7.40781463845881677337550771095, −6.02330768434725084642356421163, −5.69855232310787821554265810944, −4.12279073899064118833294374181, −2.86318575372037098407546745716, −1.76504517635723664819512494572, 0.791813146089111533954376496535, 2.85081841204981370124193443921, 3.92551257426943020964451243830, 4.60268314537499971077557302311, 6.25645877353358642118755307227, 6.86971045172317993627918432589, 7.79679242939452661799138591470, 9.018634300041175792371493910495, 9.804153133854257735966138560741, 10.27675049281184450931589973196

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