Properties

Label 4-8085e2-1.1-c1e2-0-5
Degree $4$
Conductor $65367225$
Sign $1$
Analytic cond. $4167.87$
Root an. cond. $8.03486$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s − 4-s + 2·5-s + 3·9-s − 2·11-s + 2·12-s − 4·13-s − 4·15-s − 3·16-s − 4·19-s − 2·20-s + 3·25-s − 4·27-s + 8·31-s + 4·33-s − 3·36-s + 4·37-s + 8·39-s + 4·43-s + 2·44-s + 6·45-s + 6·48-s + 4·52-s − 12·53-s − 4·55-s + 8·57-s + 4·60-s + ⋯
L(s)  = 1  − 1.15·3-s − 1/2·4-s + 0.894·5-s + 9-s − 0.603·11-s + 0.577·12-s − 1.10·13-s − 1.03·15-s − 3/4·16-s − 0.917·19-s − 0.447·20-s + 3/5·25-s − 0.769·27-s + 1.43·31-s + 0.696·33-s − 1/2·36-s + 0.657·37-s + 1.28·39-s + 0.609·43-s + 0.301·44-s + 0.894·45-s + 0.866·48-s + 0.554·52-s − 1.64·53-s − 0.539·55-s + 1.05·57-s + 0.516·60-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 65367225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 65367225 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(65367225\)    =    \(3^{2} \cdot 5^{2} \cdot 7^{4} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(4167.87\)
Root analytic conductor: \(8.03486\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{8085} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 65367225,\ (\ :1/2, 1/2),\ 1)\)

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)$
bad3$C_1$ \( ( 1 + T )^{2} \)
5$C_1$ \( ( 1 - T )^{2} \)
7 \( 1 \)
11$C_1$ \( ( 1 + T )^{2} \)
good2$C_2^2$ \( 1 + T^{2} + p^{2} T^{4} \)
13$D_{4}$ \( 1 + 4 T + 18 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
17$C_2$ \( ( 1 + p T^{2} )^{2} \)
19$D_{4}$ \( 1 + 4 T + 30 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
23$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
29$C_2^2$ \( 1 + 46 T^{2} + p^{2} T^{4} \)
31$D_{4}$ \( 1 - 8 T + 30 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 - 4 T + 30 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
41$C_2^2$ \( 1 + 70 T^{2} + p^{2} T^{4} \)
43$D_{4}$ \( 1 - 4 T + 42 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
47$C_2^2$ \( 1 + 46 T^{2} + p^{2} T^{4} \)
53$D_{4}$ \( 1 + 12 T + 94 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 + 70 T^{2} + p^{2} T^{4} \)
61$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
71$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \)
73$D_{4}$ \( 1 + 4 T + 42 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 + 20 T + 246 T^{2} + 20 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 + 24 T + 298 T^{2} + 24 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 - 12 T + 166 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
97$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
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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

−7.52073400680925859849123698071, −7.31434851072125990058655147365, −6.87374937898751580165137457252, −6.43306364105952142912263435082, −6.24981023799973078778279811393, −6.07962547639094169039755958500, −5.43592738551795314650104890813, −5.18780574120706586529271466685, −4.96161000630319140502376325118, −4.59204945088486997285361374960, −4.18498602597056118879939397501, −4.03647424449501851082716252829, −3.21719896217484920752980483116, −2.68533595096359516566716104769, −2.48716381127351422689302324811, −2.03621776118163978977057535865, −1.39498645130528337531736069637, −0.958332675761092210740496234709, 0, 0, 0.958332675761092210740496234709, 1.39498645130528337531736069637, 2.03621776118163978977057535865, 2.48716381127351422689302324811, 2.68533595096359516566716104769, 3.21719896217484920752980483116, 4.03647424449501851082716252829, 4.18498602597056118879939397501, 4.59204945088486997285361374960, 4.96161000630319140502376325118, 5.18780574120706586529271466685, 5.43592738551795314650104890813, 6.07962547639094169039755958500, 6.24981023799973078778279811393, 6.43306364105952142912263435082, 6.87374937898751580165137457252, 7.31434851072125990058655147365, 7.52073400680925859849123698071

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