Properties

Label 2-75712-1.1-c1-0-51
Degree $2$
Conductor $75712$
Sign $-1$
Analytic cond. $604.563$
Root an. cond. $24.5878$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s + 5-s − 7-s + 9-s + 2·11-s − 2·15-s − 3·17-s − 8·19-s + 2·21-s + 6·23-s − 4·25-s + 4·27-s + 9·29-s + 6·31-s − 4·33-s − 35-s + 3·37-s + 3·41-s + 2·43-s + 45-s + 12·47-s + 49-s + 6·51-s + 5·53-s + 2·55-s + 16·57-s − 14·59-s + ⋯
L(s)  = 1  − 1.15·3-s + 0.447·5-s − 0.377·7-s + 1/3·9-s + 0.603·11-s − 0.516·15-s − 0.727·17-s − 1.83·19-s + 0.436·21-s + 1.25·23-s − 4/5·25-s + 0.769·27-s + 1.67·29-s + 1.07·31-s − 0.696·33-s − 0.169·35-s + 0.493·37-s + 0.468·41-s + 0.304·43-s + 0.149·45-s + 1.75·47-s + 1/7·49-s + 0.840·51-s + 0.686·53-s + 0.269·55-s + 2.11·57-s − 1.82·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 75712 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75712 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(75712\)    =    \(2^{6} \cdot 7 \cdot 13^{2}\)
Sign: $-1$
Analytic conductor: \(604.563\)
Root analytic conductor: \(24.5878\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 75712,\ (\ :1/2),\ -1)\)

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 \( 1 \)
7 \( 1 + T \)
13 \( 1 \)
good3 \( 1 + 2 T + p T^{2} \)
5 \( 1 - T + p T^{2} \)
11 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 3 T + p T^{2} \)
19 \( 1 + 8 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 - 9 T + p T^{2} \)
31 \( 1 - 6 T + p T^{2} \)
37 \( 1 - 3 T + p T^{2} \)
41 \( 1 - 3 T + p T^{2} \)
43 \( 1 - 2 T + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 - 5 T + p T^{2} \)
59 \( 1 + 14 T + p T^{2} \)
61 \( 1 - 11 T + p T^{2} \)
67 \( 1 + 2 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 + 9 T + p T^{2} \)
79 \( 1 + 10 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
97 \( 1 + 10 T + p T^{2} \)
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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

−14.22977917472583, −13.78015887738656, −13.17889016258021, −12.79472685503285, −12.22154814856270, −11.84807287165005, −11.29399838066097, −10.79452488291532, −10.37934692350723, −9.996392374555524, −9.184214589319176, −8.768492852060837, −8.402625434851217, −7.467545265032277, −6.869804057422858, −6.398692257273034, −6.108212153528336, −5.630520982784693, −4.786138835601899, −4.416568229822763, −3.923275851797419, −2.706535946963284, −2.584434422100752, −1.470663409522489, −0.8152894624559505, 0, 0.8152894624559505, 1.470663409522489, 2.584434422100752, 2.706535946963284, 3.923275851797419, 4.416568229822763, 4.786138835601899, 5.630520982784693, 6.108212153528336, 6.398692257273034, 6.869804057422858, 7.467545265032277, 8.402625434851217, 8.768492852060837, 9.184214589319176, 9.996392374555524, 10.37934692350723, 10.79452488291532, 11.29399838066097, 11.84807287165005, 12.22154814856270, 12.79472685503285, 13.17889016258021, 13.78015887738656, 14.22977917472583

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