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Properties

Label 2-592-148.103-c1-0-13
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

\(L(1)\) \(\approx\) \(0.165045 - 0.196263i\)
\(L(\frac12)\) \(\approx\) \(0.165045 - 0.196263i\)
\(L(\frac{3}{2})\) 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

−10.29580401035806448457983467065, −9.987557843010312499110569080068, −8.789809740790038233424542194654, −7.932730751317637682335403253088, −6.68338712389667153157229057291, −5.42568106885840928728927817423, −5.02690065033765439395110203074, −3.94839348798959091080709890341, −2.68363878623416715601513671295, −0.15095747689254567803117532980, 1.65275772277886044716696466326, 2.84866434974129681409572001835, 4.60961377391060500730837515841, 5.59340218959797584494036747436, 6.66023575641722316956361749258, 7.09452167066254191074786205669, 8.085652796120921256841180095397, 9.047521512987887896821049940031, 10.31165294198597304672763995265, 11.10210336526992346244379082658

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