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Properties

Label 2-70560-1.1-c1-0-17
Degree $2$
Conductor $70560$
Sign $1$
Analytic cond. $563.424$
Root an. cond. $23.7365$
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  + 5-s + 2·13-s − 6·17-s − 4·23-s + 25-s + 2·29-s − 8·31-s + 6·37-s − 6·41-s − 12·43-s + 12·47-s + 10·53-s − 8·59-s + 10·61-s + 2·65-s + 12·67-s + 8·71-s − 10·73-s − 16·79-s − 12·83-s − 6·85-s − 6·89-s − 18·97-s + 101-s + 103-s + 107-s + 109-s + ⋯
L(s)  = 1  + 0.447·5-s + 0.554·13-s − 1.45·17-s − 0.834·23-s + 1/5·25-s + 0.371·29-s − 1.43·31-s + 0.986·37-s − 0.937·41-s − 1.82·43-s + 1.75·47-s + 1.37·53-s − 1.04·59-s + 1.28·61-s + 0.248·65-s + 1.46·67-s + 0.949·71-s − 1.17·73-s − 1.80·79-s − 1.31·83-s − 0.650·85-s − 0.635·89-s − 1.82·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 70560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 70560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(70560\)    =    \(2^{5} \cdot 3^{2} \cdot 5 \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(563.424\)
Root analytic conductor: \(23.7365\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 70560,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.780304385\)
\(L(\frac12)\) \(\approx\) \(1.780304385\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
7 \( 1 \)
good11 \( 1 + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 12 T + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 + 8 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + 18 T + p T^{2} \)
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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

−14.12073848228345, −13.56141156103538, −13.18105076082618, −12.77938194770044, −12.14096175845298, −11.46470352092966, −11.24030503884278, −10.55968182951391, −10.11450712620582, −9.597391476228556, −8.994879728085732, −8.493525657457587, −8.210597201350816, −7.201746634147262, −6.961942659592037, −6.321117809838988, −5.723408268799032, −5.336616979666936, −4.479422993047162, −4.076063123723330, −3.413552933438380, −2.603312322164783, −2.047918189452926, −1.437465114337854, −0.4329027536855103, 0.4329027536855103, 1.437465114337854, 2.047918189452926, 2.603312322164783, 3.413552933438380, 4.076063123723330, 4.479422993047162, 5.336616979666936, 5.723408268799032, 6.321117809838988, 6.961942659592037, 7.201746634147262, 8.210597201350816, 8.493525657457587, 8.994879728085732, 9.597391476228556, 10.11450712620582, 10.55968182951391, 11.24030503884278, 11.46470352092966, 12.14096175845298, 12.77938194770044, 13.18105076082618, 13.56141156103538, 14.12073848228345

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