Properties

Label 2-235340-1.1-c1-0-7
Degree $2$
Conductor $235340$
Sign $-1$
Analytic cond. $1879.19$
Root an. cond. $43.3497$
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  − 3·3-s − 5-s + 7-s + 6·9-s + 5·11-s + 3·13-s + 3·15-s + 17-s − 6·19-s − 3·21-s + 6·23-s + 25-s − 9·27-s + 9·29-s − 4·31-s − 15·33-s − 35-s + 2·37-s − 9·39-s + 10·43-s − 6·45-s + 47-s + 49-s − 3·51-s − 4·53-s − 5·55-s + 18·57-s + ⋯
L(s)  = 1  − 1.73·3-s − 0.447·5-s + 0.377·7-s + 2·9-s + 1.50·11-s + 0.832·13-s + 0.774·15-s + 0.242·17-s − 1.37·19-s − 0.654·21-s + 1.25·23-s + 1/5·25-s − 1.73·27-s + 1.67·29-s − 0.718·31-s − 2.61·33-s − 0.169·35-s + 0.328·37-s − 1.44·39-s + 1.52·43-s − 0.894·45-s + 0.145·47-s + 1/7·49-s − 0.420·51-s − 0.549·53-s − 0.674·55-s + 2.38·57-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 235340 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 235340 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(235340\)    =    \(2^{2} \cdot 5 \cdot 7 \cdot 41^{2}\)
Sign: $-1$
Analytic conductor: \(1879.19\)
Root analytic conductor: \(43.3497\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{235340} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 235340,\ (\ :1/2),\ -1)\)

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 \( 1 + T \)
7 \( 1 - T \)
41 \( 1 \)
good3 \( 1 + p T + p T^{2} \)
11 \( 1 - 5 T + p T^{2} \)
13 \( 1 - 3 T + p T^{2} \)
17 \( 1 - T + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 - 9 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
43 \( 1 - 10 T + p T^{2} \)
47 \( 1 - T + p T^{2} \)
53 \( 1 + 4 T + p T^{2} \)
59 \( 1 + 8 T + p T^{2} \)
61 \( 1 + 8 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + 13 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 + 4 T + p T^{2} \)
97 \( 1 - 13 T + p T^{2} \)
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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

−12.76608595924688, −12.63568251270341, −12.14832717272053, −11.58898598942226, −11.40988036392205, −10.87789815660855, −10.57666934563294, −10.16974500310615, −9.374106269285150, −8.776310514192385, −8.741382435174533, −7.774057947598809, −7.341602116446857, −6.803490048332904, −6.340482343216702, −6.066886808440780, −5.606003825731065, −4.746480717230695, −4.500575342151048, −4.161472852571548, −3.449315088487469, −2.780344044935943, −1.738355797465355, −1.243571414013378, −0.8376811141666450, 0, 0.8376811141666450, 1.243571414013378, 1.738355797465355, 2.780344044935943, 3.449315088487469, 4.161472852571548, 4.500575342151048, 4.746480717230695, 5.606003825731065, 6.066886808440780, 6.340482343216702, 6.803490048332904, 7.341602116446857, 7.774057947598809, 8.741382435174533, 8.776310514192385, 9.374106269285150, 10.16974500310615, 10.57666934563294, 10.87789815660855, 11.40988036392205, 11.58898598942226, 12.14832717272053, 12.63568251270341, 12.76608595924688

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