Properties

Label 4-1470e2-1.1-c1e2-0-28
Degree $4$
Conductor $2160900$
Sign $1$
Analytic cond. $137.780$
Root an. cond. $3.42607$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 3-s + 5-s + 6-s + 8-s − 10-s − 2·11-s − 4·13-s − 15-s − 16-s − 4·17-s + 2·22-s − 8·23-s − 24-s + 4·26-s + 27-s + 30-s − 2·31-s + 2·33-s + 4·34-s − 8·37-s + 4·39-s + 40-s + 4·41-s − 4·43-s + 8·46-s + 10·47-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 0.447·5-s + 0.408·6-s + 0.353·8-s − 0.316·10-s − 0.603·11-s − 1.10·13-s − 0.258·15-s − 1/4·16-s − 0.970·17-s + 0.426·22-s − 1.66·23-s − 0.204·24-s + 0.784·26-s + 0.192·27-s + 0.182·30-s − 0.359·31-s + 0.348·33-s + 0.685·34-s − 1.31·37-s + 0.640·39-s + 0.158·40-s + 0.624·41-s − 0.609·43-s + 1.17·46-s + 1.45·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2160900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(2160900\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(137.780\)
Root analytic conductor: \(3.42607\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 2160900,\ (\ :1/2, 1/2),\ 1)\)

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$$\Gal(F_p)$$F_p(T)$
bad2$C_2$ \( 1 + T + T^{2} \)
3$C_2$ \( 1 + T + T^{2} \)
5$C_2$ \( 1 - T + T^{2} \)
7 \( 1 \)
good11$C_2^2$ \( 1 + 2 T - 7 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
17$C_2^2$ \( 1 + 4 T - T^{2} + 4 p T^{3} + p^{2} T^{4} \)
19$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
23$C_2^2$ \( 1 + 8 T + 41 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + p T^{2} )^{2} \)
31$C_2^2$ \( 1 + 2 T - 27 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
37$C_2^2$ \( 1 + 8 T + 27 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
41$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 - 10 T + 53 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 2 T - 49 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 - 4 T - 43 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 + 10 T + 39 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 + 2 T - 63 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 17 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
79$C_2^2$ \( 1 + 16 T + 177 T^{2} + 16 p T^{3} + p^{2} T^{4} \)
83$C_2$ \( ( 1 + 16 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 - 14 T + 107 T^{2} - 14 p T^{3} + p^{2} T^{4} \)
97$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.305626621136665420481177327101, −8.976722965672782094092756136461, −8.515144259706806443750926299814, −8.218382500878051077703622521553, −7.61499984250085776453189081345, −7.42539388370407839818300542801, −6.79008482801315225156263092337, −6.69033163765125261491709463940, −5.79373078157689272948009113924, −5.72008765589153503044798420460, −5.29069701566200015890909289391, −4.69879211144352595803550176956, −4.16700667006997678338695722575, −3.99334699084575686077331835661, −2.80464691426811486455528942361, −2.69860925280511667535187467761, −1.88275049510877697639334908626, −1.42474143435581409294951597710, 0, 0, 1.42474143435581409294951597710, 1.88275049510877697639334908626, 2.69860925280511667535187467761, 2.80464691426811486455528942361, 3.99334699084575686077331835661, 4.16700667006997678338695722575, 4.69879211144352595803550176956, 5.29069701566200015890909289391, 5.72008765589153503044798420460, 5.79373078157689272948009113924, 6.69033163765125261491709463940, 6.79008482801315225156263092337, 7.42539388370407839818300542801, 7.61499984250085776453189081345, 8.218382500878051077703622521553, 8.515144259706806443750926299814, 8.976722965672782094092756136461, 9.305626621136665420481177327101

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