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Properties

Label 2-304200-1.1-c1-0-73
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4·7-s + 4·11-s + 6·17-s − 4·23-s + 6·29-s + 8·31-s − 2·37-s + 10·41-s + 4·43-s − 8·47-s + 9·49-s − 2·53-s + 4·59-s + 14·61-s − 12·67-s − 8·71-s − 10·73-s + 16·77-s + 4·83-s + 10·89-s − 2·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + 24·119-s + ⋯
L(s)  = 1  + 1.51·7-s + 1.20·11-s + 1.45·17-s − 0.834·23-s + 1.11·29-s + 1.43·31-s − 0.328·37-s + 1.56·41-s + 0.609·43-s − 1.16·47-s + 9/7·49-s − 0.274·53-s + 0.520·59-s + 1.79·61-s − 1.46·67-s − 0.949·71-s − 1.17·73-s + 1.82·77-s + 0.439·83-s + 1.05·89-s − 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + 2.20·119-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: \(0\)
Selberg data: \((2,\ 304200,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(5.154773175\)
\(L(\frac12)\) \(\approx\) \(5.154773175\)
\(L(\frac{3}{2})\) 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 \( 1 - 6 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
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 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 14 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
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

−12.51910345153896, −12.03101648023234, −11.76896788887594, −11.50385039985197, −10.88934326802151, −10.33773332628372, −9.979350396298792, −9.550031941411731, −8.760838730241670, −8.621164691396469, −7.958578996145137, −7.689636620832868, −7.228378800844449, −6.360476652110433, −6.257457205864259, −5.516612295922481, −5.070206465937423, −4.502804487385193, −4.132014781081714, −3.611492017470885, −2.849196691912467, −2.372086042290286, −1.481655120856879, −1.293906559994449, −0.6496012918628766, 0.6496012918628766, 1.293906559994449, 1.481655120856879, 2.372086042290286, 2.849196691912467, 3.611492017470885, 4.132014781081714, 4.502804487385193, 5.070206465937423, 5.516612295922481, 6.257457205864259, 6.360476652110433, 7.228378800844449, 7.689636620832868, 7.958578996145137, 8.621164691396469, 8.760838730241670, 9.550031941411731, 9.979350396298792, 10.33773332628372, 10.88934326802151, 11.50385039985197, 11.76896788887594, 12.03101648023234, 12.51910345153896

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