Properties

Label 2-344760-1.1-c1-0-38
Degree $2$
Conductor $344760$
Sign $-1$
Analytic cond. $2752.92$
Root an. cond. $52.4682$
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  + 3-s + 5-s + 9-s − 4·11-s + 15-s + 17-s − 8·19-s + 25-s + 27-s − 6·29-s + 8·31-s − 4·33-s − 2·37-s + 10·41-s + 4·43-s + 45-s − 7·49-s + 51-s − 6·53-s − 4·55-s − 8·57-s − 6·61-s − 4·67-s + 8·71-s + 14·73-s + 75-s + 4·79-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.447·5-s + 1/3·9-s − 1.20·11-s + 0.258·15-s + 0.242·17-s − 1.83·19-s + 1/5·25-s + 0.192·27-s − 1.11·29-s + 1.43·31-s − 0.696·33-s − 0.328·37-s + 1.56·41-s + 0.609·43-s + 0.149·45-s − 49-s + 0.140·51-s − 0.824·53-s − 0.539·55-s − 1.05·57-s − 0.768·61-s − 0.488·67-s + 0.949·71-s + 1.63·73-s + 0.115·75-s + 0.450·79-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 344760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 344760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(344760\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 13^{2} \cdot 17\)
Sign: $-1$
Analytic conductor: \(2752.92\)
Root analytic conductor: \(52.4682\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{344760} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 344760,\ (\ :1/2),\ -1)\)

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 \( 1 \)
3 \( 1 - T \)
5 \( 1 - T \)
13 \( 1 \)
17 \( 1 - T \)
good7 \( 1 + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
19 \( 1 + 8 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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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

−12.83533452170063, −12.45434043437728, −12.09960890872893, −11.14224629030060, −10.98516412707153, −10.51441660585779, −10.07260066792613, −9.570685438836916, −9.171498938623802, −8.680629612578470, −8.114030437241220, −7.835791641883099, −7.439279744691375, −6.665809946501140, −6.258805735794438, −5.934655341426149, −5.089648796802593, −4.896841749684501, −4.196189802857479, −3.747642250385112, −3.057696661360445, −2.499477657613006, −2.208919768482145, −1.583893144416419, −0.7536700505436393, 0, 0.7536700505436393, 1.583893144416419, 2.208919768482145, 2.499477657613006, 3.057696661360445, 3.747642250385112, 4.196189802857479, 4.896841749684501, 5.089648796802593, 5.934655341426149, 6.258805735794438, 6.665809946501140, 7.439279744691375, 7.835791641883099, 8.114030437241220, 8.680629612578470, 9.171498938623802, 9.570685438836916, 10.07260066792613, 10.51441660585779, 10.98516412707153, 11.14224629030060, 12.09960890872893, 12.45434043437728, 12.83533452170063

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