Properties

Label 2-2576-1.1-c1-0-19
Degree $2$
Conductor $2576$
Sign $1$
Analytic cond. $20.5694$
Root an. cond. $4.53535$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.40·3-s − 3.17·5-s − 7-s + 2.77·9-s + 1.28·11-s − 3.58·13-s − 7.63·15-s − 1.17·17-s + 8.64·19-s − 2.40·21-s + 23-s + 5.09·25-s − 0.531·27-s + 0.220·29-s + 5.94·31-s + 3.08·33-s + 3.17·35-s + 5.78·37-s − 8.60·39-s + 11.8·41-s + 8.91·43-s − 8.82·45-s + 3.77·47-s + 49-s − 2.82·51-s + 6·53-s − 4.07·55-s + ⋯
L(s)  = 1  + 1.38·3-s − 1.42·5-s − 0.377·7-s + 0.926·9-s + 0.386·11-s − 0.993·13-s − 1.97·15-s − 0.285·17-s + 1.98·19-s − 0.524·21-s + 0.208·23-s + 1.01·25-s − 0.102·27-s + 0.0410·29-s + 1.06·31-s + 0.537·33-s + 0.536·35-s + 0.951·37-s − 1.37·39-s + 1.85·41-s + 1.36·43-s − 1.31·45-s + 0.551·47-s + 0.142·49-s − 0.396·51-s + 0.824·53-s − 0.549·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2576\)    =    \(2^{4} \cdot 7 \cdot 23\)
Sign: $1$
Analytic conductor: \(20.5694\)
Root analytic conductor: \(4.53535\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2576,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.155464846\)
\(L(\frac12)\) \(\approx\) \(2.155464846\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 + T \)
23 \( 1 - T \)
good3 \( 1 - 2.40T + 3T^{2} \)
5 \( 1 + 3.17T + 5T^{2} \)
11 \( 1 - 1.28T + 11T^{2} \)
13 \( 1 + 3.58T + 13T^{2} \)
17 \( 1 + 1.17T + 17T^{2} \)
19 \( 1 - 8.64T + 19T^{2} \)
29 \( 1 - 0.220T + 29T^{2} \)
31 \( 1 - 5.94T + 31T^{2} \)
37 \( 1 - 5.78T + 37T^{2} \)
41 \( 1 - 11.8T + 41T^{2} \)
43 \( 1 - 8.91T + 43T^{2} \)
47 \( 1 - 3.77T + 47T^{2} \)
53 \( 1 - 6T + 53T^{2} \)
59 \( 1 + 5.28T + 59T^{2} \)
61 \( 1 - 5.74T + 61T^{2} \)
67 \( 1 + 9.46T + 67T^{2} \)
71 \( 1 + 14.8T + 71T^{2} \)
73 \( 1 - 2.49T + 73T^{2} \)
79 \( 1 - 13.4T + 79T^{2} \)
83 \( 1 + 1.26T + 83T^{2} \)
89 \( 1 + 9.35T + 89T^{2} \)
97 \( 1 - 8.14T + 97T^{2} \)
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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

−8.983699208571697966929522675432, −7.899073287547917677027616274251, −7.62792398370686183290551296227, −7.04278933426638651697996257436, −5.79282701832457550817484280738, −4.56259199728125820685708404624, −3.96747059972615061942194420608, −3.08527301204951525885525314895, −2.55424495130491995037081821154, −0.870377617698484628462871288234, 0.870377617698484628462871288234, 2.55424495130491995037081821154, 3.08527301204951525885525314895, 3.96747059972615061942194420608, 4.56259199728125820685708404624, 5.79282701832457550817484280738, 7.04278933426638651697996257436, 7.62792398370686183290551296227, 7.899073287547917677027616274251, 8.983699208571697966929522675432

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