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Properties

Label 2-304200-1.1-c1-0-51
Degree $2$
Conductor $304200$
Sign $-1$
Analytic cond. $2429.04$
Root an. cond. $49.2853$
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  − 4·7-s − 4·11-s − 8·17-s + 6·23-s + 8·29-s + 2·31-s + 4·37-s + 10·41-s − 4·43-s − 8·47-s + 9·49-s − 10·53-s − 12·59-s − 2·61-s + 2·67-s + 12·71-s + 2·73-s + 16·77-s + 8·79-s − 8·83-s + 6·89-s + 10·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯
L(s)  = 1  − 1.51·7-s − 1.20·11-s − 1.94·17-s + 1.25·23-s + 1.48·29-s + 0.359·31-s + 0.657·37-s + 1.56·41-s − 0.609·43-s − 1.16·47-s + 9/7·49-s − 1.37·53-s − 1.56·59-s − 0.256·61-s + 0.244·67-s + 1.42·71-s + 0.234·73-s + 1.82·77-s + 0.900·79-s − 0.878·83-s + 0.635·89-s + 1.01·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 304200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(304200\)    =    \(2^{3} \cdot 3^{2} \cdot 5^{2} \cdot 13^{2}\)
Sign: $-1$
Analytic conductor: \(2429.04\)
Root analytic conductor: \(49.2853\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 304200,\ (\ :1/2),\ -1)\)

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 \( 1 \)
13 \( 1 \)
good7 \( 1 + 4 T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
17 \( 1 + 8 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 - 8 T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 - 4 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 8 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
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

−13.00947844728908, −12.60744445287002, −12.14938757782905, −11.36034332229572, −11.07922601475128, −10.56176628203328, −10.27526917131283, −9.480307765830197, −9.399206037094242, −8.884134241832295, −8.196920484835505, −7.917131227915159, −7.219663454807321, −6.675749662405775, −6.408610454726444, −6.078823933842709, −5.225794540359698, −4.780603554093925, −4.441403580425668, −3.657062245164416, −3.094641081833077, −2.635901654696170, −2.376138783887021, −1.377163828931421, −0.5617071903463184, 0, 0.5617071903463184, 1.377163828931421, 2.376138783887021, 2.635901654696170, 3.094641081833077, 3.657062245164416, 4.441403580425668, 4.780603554093925, 5.225794540359698, 6.078823933842709, 6.408610454726444, 6.675749662405775, 7.219663454807321, 7.917131227915159, 8.196920484835505, 8.884134241832295, 9.399206037094242, 9.480307765830197, 10.27526917131283, 10.56176628203328, 11.07922601475128, 11.36034332229572, 12.14938757782905, 12.60744445287002, 13.00947844728908

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