Properties

Label 4-8470e2-1.1-c1e2-0-5
Degree $4$
Conductor $71740900$
Sign $1$
Analytic cond. $4574.26$
Root an. cond. $8.22394$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + 2·3-s + 3·4-s − 2·5-s − 4·6-s + 2·7-s − 4·8-s + 4·10-s + 6·12-s + 8·13-s − 4·14-s − 4·15-s + 5·16-s + 8·17-s + 2·19-s − 6·20-s + 4·21-s + 6·23-s − 8·24-s + 3·25-s − 16·26-s − 2·27-s + 6·28-s + 2·29-s + 8·30-s + 8·31-s − 6·32-s + ⋯
L(s)  = 1  − 1.41·2-s + 1.15·3-s + 3/2·4-s − 0.894·5-s − 1.63·6-s + 0.755·7-s − 1.41·8-s + 1.26·10-s + 1.73·12-s + 2.21·13-s − 1.06·14-s − 1.03·15-s + 5/4·16-s + 1.94·17-s + 0.458·19-s − 1.34·20-s + 0.872·21-s + 1.25·23-s − 1.63·24-s + 3/5·25-s − 3.13·26-s − 0.384·27-s + 1.13·28-s + 0.371·29-s + 1.46·30-s + 1.43·31-s − 1.06·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 71740900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 71740900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(71740900\)    =    \(2^{2} \cdot 5^{2} \cdot 7^{2} \cdot 11^{4}\)
Sign: $1$
Analytic conductor: \(4574.26\)
Root analytic conductor: \(8.22394\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{8470} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 71740900,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.011797545\)
\(L(\frac12)\) \(\approx\) \(4.011797545\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 + T )^{2} \)
5$C_1$ \( ( 1 + T )^{2} \)
7$C_1$ \( ( 1 - T )^{2} \)
11 \( 1 \)
good3$D_{4}$ \( 1 - 2 T + 4 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
17$D_{4}$ \( 1 - 8 T + 38 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 - 2 T + 36 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
23$D_{4}$ \( 1 - 6 T + 28 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 - 2 T - 16 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 - 8 T + 66 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 + 2 T + 72 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 + 6 T + 64 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 - 4 T + 42 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
47$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
53$D_{4}$ \( 1 - 14 T + 152 T^{2} - 14 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 + 4 T + 14 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
61$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
67$D_{4}$ \( 1 - 4 T - 54 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 - 16 T + 158 T^{2} - 16 p T^{3} + p^{2} T^{4} \)
73$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
79$D_{4}$ \( 1 + 2 T + 84 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 - 16 T + 182 T^{2} - 16 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 + 4 T + 134 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 - 2 T + 192 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
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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

−7.998230116829811816063117787802, −7.948602980300326526203846525177, −7.30971673960683305591602723172, −7.27458211560247622726114967852, −6.85472159487226386161585787760, −6.31944989851239131294642289650, −5.88869373553919900668073849168, −5.81017465198320629073105775471, −5.15990683524937834056175508296, −4.89671794435961244188185087299, −4.26330379660979631485177717867, −3.83080123405618739694916607871, −3.46777737609915693610429276406, −3.26931317018699918414215610111, −2.75469352950796797406717442898, −2.58948697244380533346049524957, −1.79826922249000786052941595094, −1.33382075137780746362590914466, −0.904022675554635204604662048525, −0.74346325241763926075969472272, 0.74346325241763926075969472272, 0.904022675554635204604662048525, 1.33382075137780746362590914466, 1.79826922249000786052941595094, 2.58948697244380533346049524957, 2.75469352950796797406717442898, 3.26931317018699918414215610111, 3.46777737609915693610429276406, 3.83080123405618739694916607871, 4.26330379660979631485177717867, 4.89671794435961244188185087299, 5.15990683524937834056175508296, 5.81017465198320629073105775471, 5.88869373553919900668073849168, 6.31944989851239131294642289650, 6.85472159487226386161585787760, 7.27458211560247622726114967852, 7.30971673960683305591602723172, 7.948602980300326526203846525177, 7.998230116829811816063117787802

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