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Properties

Label 2-177870-1.1-c1-0-128
Degree $2$
Conductor $177870$
Sign $-1$
Analytic cond. $1420.29$
Root an. cond. $37.6868$
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  + 2-s − 3-s + 4-s − 5-s − 6-s + 8-s + 9-s − 10-s − 12-s + 15-s + 16-s − 5·17-s + 18-s − 6·19-s − 20-s − 24-s + 25-s − 27-s + 6·29-s + 30-s − 5·31-s + 32-s − 5·34-s + 36-s + 4·37-s − 6·38-s − 40-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s − 0.408·6-s + 0.353·8-s + 1/3·9-s − 0.316·10-s − 0.288·12-s + 0.258·15-s + 1/4·16-s − 1.21·17-s + 0.235·18-s − 1.37·19-s − 0.223·20-s − 0.204·24-s + 1/5·25-s − 0.192·27-s + 1.11·29-s + 0.182·30-s − 0.898·31-s + 0.176·32-s − 0.857·34-s + 1/6·36-s + 0.657·37-s − 0.973·38-s − 0.158·40-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177870 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177870 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177870\)    =    \(2 \cdot 3 \cdot 5 \cdot 7^{2} \cdot 11^{2}\)
Sign: $-1$
Analytic conductor: \(1420.29\)
Root analytic conductor: \(37.6868\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 177870,\ (\ :1/2),\ -1)\)

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 \( 1 - T \)
3 \( 1 + T \)
5 \( 1 + T \)
7 \( 1 \)
11 \( 1 \)
good13 \( 1 + p T^{2} \)
17 \( 1 + 5 T + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 5 T + p T^{2} \)
37 \( 1 - 4 T + p T^{2} \)
41 \( 1 + 7 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 7 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 - 8 T + p T^{2} \)
67 \( 1 - 14 T + p T^{2} \)
71 \( 1 + 11 T + p T^{2} \)
73 \( 1 + 13 T + p T^{2} \)
79 \( 1 + 5 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + 3 T + p T^{2} \)
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

−13.28447303416379, −13.02081505907536, −12.30806667654037, −12.20733615560334, −11.43459845257301, −11.22811667234903, −10.70707486209947, −10.28342166191647, −9.800303460432830, −9.010597159680889, −8.518748207341808, −8.256789459618622, −7.376398647507335, −6.977434783834488, −6.647819517013746, −5.996820637257229, −5.672497801221874, −4.848665355625938, −4.551293054869308, −4.096891921626406, −3.540103220675595, −2.808081964491019, −2.209280487391716, −1.666930265966275, −0.7298981786882132, 0, 0.7298981786882132, 1.666930265966275, 2.209280487391716, 2.808081964491019, 3.540103220675595, 4.096891921626406, 4.551293054869308, 4.848665355625938, 5.672497801221874, 5.996820637257229, 6.647819517013746, 6.977434783834488, 7.376398647507335, 8.256789459618622, 8.518748207341808, 9.010597159680889, 9.800303460432830, 10.28342166191647, 10.70707486209947, 11.22811667234903, 11.43459845257301, 12.20733615560334, 12.30806667654037, 13.02081505907536, 13.28447303416379

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