Properties

Label 2-7600-1.1-c1-0-93
Degree $2$
Conductor $7600$
Sign $1$
Analytic cond. $60.6863$
Root an. cond. $7.79014$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·3-s + 4·7-s + 9-s + 4·11-s − 6·17-s + 19-s + 8·21-s + 8·23-s − 4·27-s − 6·29-s + 8·31-s + 8·33-s + 8·37-s − 2·41-s + 12·47-s + 9·49-s − 12·51-s − 4·53-s + 2·57-s − 8·59-s − 14·61-s + 4·63-s − 2·67-s + 16·69-s + 8·71-s + 2·73-s + 16·77-s + ⋯
L(s)  = 1  + 1.15·3-s + 1.51·7-s + 1/3·9-s + 1.20·11-s − 1.45·17-s + 0.229·19-s + 1.74·21-s + 1.66·23-s − 0.769·27-s − 1.11·29-s + 1.43·31-s + 1.39·33-s + 1.31·37-s − 0.312·41-s + 1.75·47-s + 9/7·49-s − 1.68·51-s − 0.549·53-s + 0.264·57-s − 1.04·59-s − 1.79·61-s + 0.503·63-s − 0.244·67-s + 1.92·69-s + 0.949·71-s + 0.234·73-s + 1.82·77-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: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 7600,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.233443038\)
\(L(\frac12)\) \(\approx\) \(4.233443038\)
\(L(\frac{3}{2})\) 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 \( 1 - 2 T + p T^{2} \)
7 \( 1 - 4 T + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 + 4 T + p T^{2} \)
59 \( 1 + 8 T + p T^{2} \)
61 \( 1 + 14 T + p T^{2} \)
67 \( 1 + 2 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + p T^{2} \)
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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

−7.925358914978323744069750802867, −7.41087501458715986305350732750, −6.62542452791134774556625333336, −5.81608719777571159025585660718, −4.72123128961115813751033883439, −4.41245067743274871039922267955, −3.48891652123959845609956411943, −2.61042469281254666762754476140, −1.89182704007716462077500996249, −1.04872841385412282787809056704, 1.04872841385412282787809056704, 1.89182704007716462077500996249, 2.61042469281254666762754476140, 3.48891652123959845609956411943, 4.41245067743274871039922267955, 4.72123128961115813751033883439, 5.81608719777571159025585660718, 6.62542452791134774556625333336, 7.41087501458715986305350732750, 7.925358914978323744069750802867

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