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Properties

Label 2-592-37.26-c1-0-7
Degree $2$
Conductor $592$
Sign $0.471 - 0.882i$
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  + (−1.08 + 1.88i)3-s + (0.5 − 0.866i)5-s + (0.370 − 0.641i)7-s + (−0.870 − 1.50i)9-s + 2.91·11-s + (1.54 − 2.68i)13-s + (1.08 + 1.88i)15-s + (3.41 + 5.92i)17-s + (0.0887 − 0.153i)19-s + (0.806 + 1.39i)21-s + 3.43·23-s + (2 + 3.46i)25-s − 2.74·27-s − 4.61·29-s − 2.91·31-s + ⋯
L(s)  = 1  + (−0.628 + 1.08i)3-s + (0.223 − 0.387i)5-s + (0.140 − 0.242i)7-s + (−0.290 − 0.502i)9-s + 0.879·11-s + (0.429 − 0.743i)13-s + (0.281 + 0.486i)15-s + (0.829 + 1.43i)17-s + (0.0203 − 0.0352i)19-s + (0.176 + 0.304i)21-s + 0.716·23-s + (0.400 + 0.692i)25-s − 0.527·27-s − 0.856·29-s − 0.524·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.471 - 0.882i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 592 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.471 - 0.882i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(592\)    =    \(2^{4} \cdot 37\)
Sign: $0.471 - 0.882i$
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: \((2,\ 592,\ (\ :1/2),\ 0.471 - 0.882i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.15103 + 0.690199i\)
\(L(\frac12)\) \(\approx\) \(1.15103 + 0.690199i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.04 - 5.72i)T \)
good3 \( 1 + (1.08 - 1.88i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (-0.5 + 0.866i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (-0.370 + 0.641i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 - 2.91T + 11T^{2} \)
13 \( 1 + (-1.54 + 2.68i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-3.41 - 5.92i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.0887 + 0.153i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 - 3.43T + 23T^{2} \)
29 \( 1 + 4.61T + 29T^{2} \)
31 \( 1 + 2.91T + 31T^{2} \)
41 \( 1 + (0.322 - 0.558i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 - 3.48T + 43T^{2} \)
47 \( 1 - 9.79T + 47T^{2} \)
53 \( 1 + (3.37 + 5.83i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-3.72 - 6.45i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-3.67 + 6.36i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (5.18 - 8.98i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-2.72 + 4.72i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 - 12.3T + 73T^{2} \)
79 \( 1 + (6.26 - 10.8i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (4.18 + 7.24i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (7.85 + 13.6i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 16.8T + 97T^{2} \)
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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

−10.78670166333803890351108089193, −10.07315269832823258385190217457, −9.248502184236362695774517187438, −8.398131914026157084543556913883, −7.24041032975179334035036522696, −5.93167298189212338003965530543, −5.36071449538516443739461965114, −4.24682874344543763245977686977, −3.45948987602340826279893889791, −1.34965228698998507930127492707, 1.00121999316913041046562763312, 2.29721989519170572514837296450, 3.79475800993879104745308891001, 5.25348923935079515668957889673, 6.13791545853542080047479164517, 6.96646711069549768213731149906, 7.48402505599287067297029048993, 8.878837903389863458564446671622, 9.531586131055382999876666856033, 10.81362707454883410691392517701

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