Properties

Label 2-7600-1.1-c1-0-113
Degree $2$
Conductor $7600$
Sign $-1$
Analytic cond. $60.6863$
Root an. cond. $7.79014$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.750·3-s − 0.872·7-s − 2.43·9-s + 1.11·11-s + 4.58·13-s − 2.96·17-s − 19-s + 0.654·21-s − 3.83·23-s + 4.07·27-s + 7.56·29-s − 9.56·31-s − 0.832·33-s + 0.750·37-s − 3.43·39-s + 8.87·41-s + 11.5·43-s − 8.53·47-s − 6.23·49-s + 2.22·51-s + 8.17·53-s + 0.750·57-s − 4.65·59-s + 0.889·61-s + 2.12·63-s − 2.59·67-s + 2.87·69-s + ⋯
L(s)  = 1  − 0.433·3-s − 0.329·7-s − 0.812·9-s + 0.334·11-s + 1.27·13-s − 0.717·17-s − 0.229·19-s + 0.142·21-s − 0.799·23-s + 0.784·27-s + 1.40·29-s − 1.71·31-s − 0.144·33-s + 0.123·37-s − 0.550·39-s + 1.38·41-s + 1.75·43-s − 1.24·47-s − 0.891·49-s + 0.310·51-s + 1.12·53-s + 0.0993·57-s − 0.605·59-s + 0.113·61-s + 0.268·63-s − 0.316·67-s + 0.346·69-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(7600\)    =    \(2^{4} \cdot 5^{2} \cdot 19\)
Sign: $-1$
Analytic conductor: \(60.6863\)
Root analytic conductor: \(7.79014\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 7600,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
19 \( 1 + T \)
good3 \( 1 + 0.750T + 3T^{2} \)
7 \( 1 + 0.872T + 7T^{2} \)
11 \( 1 - 1.11T + 11T^{2} \)
13 \( 1 - 4.58T + 13T^{2} \)
17 \( 1 + 2.96T + 17T^{2} \)
23 \( 1 + 3.83T + 23T^{2} \)
29 \( 1 - 7.56T + 29T^{2} \)
31 \( 1 + 9.56T + 31T^{2} \)
37 \( 1 - 0.750T + 37T^{2} \)
41 \( 1 - 8.87T + 41T^{2} \)
43 \( 1 - 11.5T + 43T^{2} \)
47 \( 1 + 8.53T + 47T^{2} \)
53 \( 1 - 8.17T + 53T^{2} \)
59 \( 1 + 4.65T + 59T^{2} \)
61 \( 1 - 0.889T + 61T^{2} \)
67 \( 1 + 2.59T + 67T^{2} \)
71 \( 1 + 7.34T + 71T^{2} \)
73 \( 1 + 11.8T + 73T^{2} \)
79 \( 1 + 4T + 79T^{2} \)
83 \( 1 - 3.83T + 83T^{2} \)
89 \( 1 - 1.34T + 89T^{2} \)
97 \( 1 - 8.17T + 97T^{2} \)
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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

−7.52514813441856358792524040022, −6.62635174300668891410521802882, −6.08341690419783228144870070077, −5.69782504690430844768493870955, −4.64392531789457378114684349092, −3.96503586459375227395186192054, −3.16143301157324559488549254145, −2.27329217694279755324967747798, −1.15164366327805649795430619188, 0, 1.15164366327805649795430619188, 2.27329217694279755324967747798, 3.16143301157324559488549254145, 3.96503586459375227395186192054, 4.64392531789457378114684349092, 5.69782504690430844768493870955, 6.08341690419783228144870070077, 6.62635174300668891410521802882, 7.52514813441856358792524040022

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