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Properties

Label 2-572-11.5-c1-0-0
Degree $2$
Conductor $572$
Sign $0.359 - 0.933i$
Analytic cond. $4.56744$
Root an. cond. $2.13715$
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  + (−2.46 − 1.79i)3-s + (0.518 − 1.59i)5-s + (−2.24 + 1.62i)7-s + (1.94 + 5.98i)9-s + (−3.14 − 1.04i)11-s + (−0.309 − 0.951i)13-s + (−4.13 + 3.00i)15-s + (−0.872 + 2.68i)17-s + (−0.580 − 0.421i)19-s + 8.44·21-s + 1.69·23-s + (1.77 + 1.28i)25-s + (3.10 − 9.55i)27-s + (−0.142 + 0.103i)29-s + (3.16 + 9.73i)31-s + ⋯
L(s)  = 1  + (−1.42 − 1.03i)3-s + (0.231 − 0.713i)5-s + (−0.847 + 0.615i)7-s + (0.648 + 1.99i)9-s + (−0.948 − 0.316i)11-s + (−0.0857 − 0.263i)13-s + (−1.06 + 0.775i)15-s + (−0.211 + 0.651i)17-s + (−0.133 − 0.0967i)19-s + 1.84·21-s + 0.354·23-s + (0.354 + 0.257i)25-s + (0.597 − 1.83i)27-s + (−0.0263 + 0.0191i)29-s + (0.568 + 1.74i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 572 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.359 - 0.933i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 572 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.359 - 0.933i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(572\)    =    \(2^{2} \cdot 11 \cdot 13\)
Sign: $0.359 - 0.933i$
Analytic conductor: \(4.56744\)
Root analytic conductor: \(2.13715\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{572} (313, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 572,\ (\ :1/2),\ 0.359 - 0.933i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.262449 + 0.180190i\)
\(L(\frac12)\) \(\approx\) \(0.262449 + 0.180190i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
11 \( 1 + (3.14 + 1.04i)T \)
13 \( 1 + (0.309 + 0.951i)T \)
good3 \( 1 + (2.46 + 1.79i)T + (0.927 + 2.85i)T^{2} \)
5 \( 1 + (-0.518 + 1.59i)T + (-4.04 - 2.93i)T^{2} \)
7 \( 1 + (2.24 - 1.62i)T + (2.16 - 6.65i)T^{2} \)
17 \( 1 + (0.872 - 2.68i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (0.580 + 0.421i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 - 1.69T + 23T^{2} \)
29 \( 1 + (0.142 - 0.103i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (-3.16 - 9.73i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (0.763 - 0.554i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (-3.83 - 2.78i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 - 4.08T + 43T^{2} \)
47 \( 1 + (1.40 + 1.02i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (3.11 + 9.59i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (12.0 - 8.76i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (4.00 - 12.3i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 - 2.13T + 67T^{2} \)
71 \( 1 + (-1.29 + 3.99i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (6.76 - 4.91i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (2.09 + 6.45i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (2.64 - 8.12i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 - 2.99T + 89T^{2} \)
97 \( 1 + (3.07 + 9.47i)T + (-78.4 + 57.0i)T^{2} \)
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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

−10.92646994547399883510923966859, −10.32457950471268695690342042541, −9.082746780616764838593769999781, −8.158611497752785203936886356934, −7.09254640317043292287080531142, −6.22189223031542426335335347834, −5.55494112788714787358820808944, −4.79447248170758551106744945515, −2.81518230434217368517156746026, −1.28208691530972227543099451649, 0.23146273818043284342149854499, 2.82663675356509203541886068446, 4.09237651515461853193215883871, 4.94800990034681953834814743737, 6.02007115572546590103671581052, 6.65434838843270332272301121609, 7.62080627153068079573242192028, 9.384113380157694514446783633112, 9.855485940380939430802184838348, 10.70981820419414067291248053932

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