Properties

Label 2-575-1.1-c3-0-84
Degree $2$
Conductor $575$
Sign $-1$
Analytic cond. $33.9260$
Root an. cond. $5.82461$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.404·2-s + 7.11·3-s − 7.83·4-s − 2.88·6-s − 13.7·7-s + 6.41·8-s + 23.5·9-s + 24.2·11-s − 55.7·12-s − 3.05·13-s + 5.58·14-s + 60.0·16-s − 63.1·17-s − 9.55·18-s − 2.07·19-s − 98.0·21-s − 9.81·22-s + 23·23-s + 45.6·24-s + 1.23·26-s − 24.1·27-s + 108.·28-s − 8.16·29-s − 156.·31-s − 75.6·32-s + 172.·33-s + 25.5·34-s + ⋯
L(s)  = 1  − 0.143·2-s + 1.36·3-s − 0.979·4-s − 0.195·6-s − 0.744·7-s + 0.283·8-s + 0.874·9-s + 0.664·11-s − 1.34·12-s − 0.0650·13-s + 0.106·14-s + 0.938·16-s − 0.900·17-s − 0.125·18-s − 0.0250·19-s − 1.01·21-s − 0.0950·22-s + 0.208·23-s + 0.387·24-s + 0.00931·26-s − 0.172·27-s + 0.729·28-s − 0.0522·29-s − 0.909·31-s − 0.417·32-s + 0.909·33-s + 0.128·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 575 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(575\)    =    \(5^{2} \cdot 23\)
Sign: $-1$
Analytic conductor: \(33.9260\)
Root analytic conductor: \(5.82461\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 575,\ (\ :3/2),\ -1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
23 \( 1 - 23T \)
good2 \( 1 + 0.404T + 8T^{2} \)
3 \( 1 - 7.11T + 27T^{2} \)
7 \( 1 + 13.7T + 343T^{2} \)
11 \( 1 - 24.2T + 1.33e3T^{2} \)
13 \( 1 + 3.05T + 2.19e3T^{2} \)
17 \( 1 + 63.1T + 4.91e3T^{2} \)
19 \( 1 + 2.07T + 6.85e3T^{2} \)
29 \( 1 + 8.16T + 2.43e4T^{2} \)
31 \( 1 + 156.T + 2.97e4T^{2} \)
37 \( 1 + 302.T + 5.06e4T^{2} \)
41 \( 1 + 42.7T + 6.89e4T^{2} \)
43 \( 1 + 215.T + 7.95e4T^{2} \)
47 \( 1 + 247.T + 1.03e5T^{2} \)
53 \( 1 + 600.T + 1.48e5T^{2} \)
59 \( 1 - 92.2T + 2.05e5T^{2} \)
61 \( 1 - 532.T + 2.26e5T^{2} \)
67 \( 1 + 30.3T + 3.00e5T^{2} \)
71 \( 1 + 736.T + 3.57e5T^{2} \)
73 \( 1 + 349.T + 3.89e5T^{2} \)
79 \( 1 - 301.T + 4.93e5T^{2} \)
83 \( 1 + 139.T + 5.71e5T^{2} \)
89 \( 1 - 859.T + 7.04e5T^{2} \)
97 \( 1 - 927.T + 9.12e5T^{2} \)
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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

−9.540831325529287068124832935839, −9.001053010521063806429468037041, −8.428338935532596917710181146798, −7.41791625572660601008505153522, −6.38608074958207755446954378696, −4.97792619297874450915628785636, −3.85499561665351579744521972204, −3.18782697327598424773988212871, −1.75626484886329951011006257511, 0, 1.75626484886329951011006257511, 3.18782697327598424773988212871, 3.85499561665351579744521972204, 4.97792619297874450915628785636, 6.38608074958207755446954378696, 7.41791625572660601008505153522, 8.428338935532596917710181146798, 9.001053010521063806429468037041, 9.540831325529287068124832935839

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