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Properties

Label 2-510e2-1.1-c1-0-52
Degree $2$
Conductor $260100$
Sign $-1$
Analytic cond. $2076.90$
Root an. cond. $45.5731$
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  − 7-s − 5·13-s − 19-s − 6·23-s − 5·31-s − 37-s − 5·43-s − 12·47-s − 6·49-s + 12·53-s − 12·59-s + 61-s + 13·67-s + 6·71-s + 2·73-s − 8·79-s − 6·83-s + 6·89-s + 5·91-s + 17·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯
L(s)  = 1  − 0.377·7-s − 1.38·13-s − 0.229·19-s − 1.25·23-s − 0.898·31-s − 0.164·37-s − 0.762·43-s − 1.75·47-s − 6/7·49-s + 1.64·53-s − 1.56·59-s + 0.128·61-s + 1.58·67-s + 0.712·71-s + 0.234·73-s − 0.900·79-s − 0.658·83-s + 0.635·89-s + 0.524·91-s + 1.72·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 260100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 260100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(260100\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 17^{2}\)
Sign: $-1$
Analytic conductor: \(2076.90\)
Root analytic conductor: \(45.5731\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 260100,\ (\ :1/2),\ -1)\)

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 \( 1 \)
5 \( 1 \)
17 \( 1 \)
good7 \( 1 + T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 + 5 T + p T^{2} \)
19 \( 1 + T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + 5 T + p T^{2} \)
37 \( 1 + T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + 5 T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 - 12 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 - T + p T^{2} \)
67 \( 1 - 13 T + p T^{2} \)
71 \( 1 - 6 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 17 T + p T^{2} \)
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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

−13.13123078803344, −12.42809809177199, −12.30580238664960, −11.59373940010012, −11.37648454904512, −10.67048085348654, −10.11366436780227, −9.889263361363646, −9.458800900674688, −8.888528138602923, −8.342079937084949, −7.880825560697635, −7.427442203368308, −6.901820675735335, −6.471627387307226, −5.936672933479283, −5.371289178422933, −4.861359980212717, −4.458323940402915, −3.690965860604269, −3.357315170677178, −2.638771959888386, −2.047943519198960, −1.665279539455500, −0.5789553831587404, 0, 0.5789553831587404, 1.665279539455500, 2.047943519198960, 2.638771959888386, 3.357315170677178, 3.690965860604269, 4.458323940402915, 4.861359980212717, 5.371289178422933, 5.936672933479283, 6.471627387307226, 6.901820675735335, 7.427442203368308, 7.880825560697635, 8.342079937084949, 8.888528138602923, 9.458800900674688, 9.889263361363646, 10.11366436780227, 10.67048085348654, 11.37648454904512, 11.59373940010012, 12.30580238664960, 12.42809809177199, 13.13123078803344

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