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Properties

Label 2-572-572.259-c1-0-43
Degree $2$
Conductor $572$
Sign $0.996 - 0.0875i$
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  + (1.34 − 0.437i)2-s + (1.61 − 1.17i)4-s + (2.59 + 3.56i)7-s + (1.66 − 2.28i)8-s + (0.927 + 2.85i)9-s + (−0.815 + 3.21i)11-s + (−3.42 + 1.11i)13-s + (5.04 + 3.66i)14-s + (1.23 − 3.80i)16-s + (−1.49 − 0.486i)17-s + (2.49 + 3.43i)18-s + (5.12 − 7.05i)19-s + (0.307 + 4.68i)22-s + (−4.04 − 2.93i)25-s + (−4.12 + 2.99i)26-s + ⋯
L(s)  = 1  + (0.951 − 0.309i)2-s + (0.809 − 0.587i)4-s + (0.980 + 1.34i)7-s + (0.587 − 0.809i)8-s + (0.309 + 0.951i)9-s + (−0.245 + 0.969i)11-s + (−0.951 + 0.309i)13-s + (1.34 + 0.980i)14-s + (0.309 − 0.951i)16-s + (−0.363 − 0.118i)17-s + (0.587 + 0.809i)18-s + (1.17 − 1.61i)19-s + (0.0656 + 0.997i)22-s + (−0.809 − 0.587i)25-s + (−0.809 + 0.587i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 572 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 - 0.0875i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 572 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.996 - 0.0875i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(572\)    =    \(2^{2} \cdot 11 \cdot 13\)
Sign: $0.996 - 0.0875i$
Analytic conductor: \(4.56744\)
Root analytic conductor: \(2.13715\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{572} (259, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 572,\ (\ :1/2),\ 0.996 - 0.0875i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.78556 + 0.122152i\)
\(L(\frac12)\) \(\approx\) \(2.78556 + 0.122152i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.34 + 0.437i)T \)
11 \( 1 + (0.815 - 3.21i)T \)
13 \( 1 + (3.42 - 1.11i)T \)
good3 \( 1 + (-0.927 - 2.85i)T^{2} \)
5 \( 1 + (4.04 + 2.93i)T^{2} \)
7 \( 1 + (-2.59 - 3.56i)T + (-2.16 + 6.65i)T^{2} \)
17 \( 1 + (1.49 + 0.486i)T + (13.7 + 9.99i)T^{2} \)
19 \( 1 + (-5.12 + 7.05i)T + (-5.87 - 18.0i)T^{2} \)
23 \( 1 - 23T^{2} \)
29 \( 1 + (0.665 + 0.915i)T + (-8.96 + 27.5i)T^{2} \)
31 \( 1 + (3.07 + 9.45i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (-11.4 + 35.1i)T^{2} \)
41 \( 1 + (12.6 + 38.9i)T^{2} \)
43 \( 1 + 43T^{2} \)
47 \( 1 + (-11.0 - 8.04i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (2.11 + 6.52i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (-9.33 + 6.78i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (14.4 + 4.69i)T + (49.3 + 35.8i)T^{2} \)
67 \( 1 + 5.09T + 67T^{2} \)
71 \( 1 + (3.53 - 10.8i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (-0.934 - 0.303i)T + (67.1 + 48.7i)T^{2} \)
89 \( 1 - 89T^{2} \)
97 \( 1 + (78.4 - 57.0i)T^{2} \)
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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

−11.15633879639628169778277654392, −9.932254267892147141588025947924, −9.219399899598323534375776358107, −7.77625108704907454218783578321, −7.22117049813488367228632705575, −5.80449143726149161645529184739, −4.95240109067000600896867795654, −4.49282745992651376300574758227, −2.51019518471020181763393126277, −2.07645569586469594137790973433, 1.42284473389365422836683897557, 3.29972740083115004159745757534, 4.02068032728153662814871938635, 5.13048852089741089789972534087, 5.99698882884037371695145639619, 7.33643495499676290168140389234, 7.55811804287876496643138879560, 8.757467692726406891914600870114, 10.18499655541604120047460038317, 10.77387090502099638879897070606

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