Properties

Label 4-315e2-1.1-c1e2-0-34
Degree $4$
Conductor $99225$
Sign $1$
Analytic cond. $6.32667$
Root an. cond. $1.58596$
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  − 2·2-s + 4-s − 2·5-s − 2·7-s + 4·10-s − 4·11-s − 4·13-s + 4·14-s + 16-s − 4·17-s − 2·20-s + 8·22-s − 4·23-s + 3·25-s + 8·26-s − 2·28-s − 16·29-s + 2·32-s + 8·34-s + 4·35-s − 12·37-s + 4·41-s − 8·43-s − 4·44-s + 8·46-s + 8·47-s + 3·49-s + ⋯
L(s)  = 1  − 1.41·2-s + 1/2·4-s − 0.894·5-s − 0.755·7-s + 1.26·10-s − 1.20·11-s − 1.10·13-s + 1.06·14-s + 1/4·16-s − 0.970·17-s − 0.447·20-s + 1.70·22-s − 0.834·23-s + 3/5·25-s + 1.56·26-s − 0.377·28-s − 2.97·29-s + 0.353·32-s + 1.37·34-s + 0.676·35-s − 1.97·37-s + 0.624·41-s − 1.21·43-s − 0.603·44-s + 1.17·46-s + 1.16·47-s + 3/7·49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 99225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99225 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(99225\)    =    \(3^{4} \cdot 5^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(6.32667\)
Root analytic conductor: \(1.58596\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 99225,\ (\ :1/2, 1/2),\ 1)\)

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 \( 1 \)
5$C_1$ \( ( 1 + T )^{2} \)
7$C_1$ \( ( 1 + T )^{2} \)
good2$D_{4}$ \( 1 + p T + 3 T^{2} + p^{2} T^{3} + p^{2} T^{4} \)
11$D_{4}$ \( 1 + 4 T + 18 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
13$D_{4}$ \( 1 + 4 T + 22 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
17$C_4$ \( 1 + 4 T + 6 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
19$C_2^2$ \( 1 + 30 T^{2} + p^{2} T^{4} \)
23$D_{4}$ \( 1 + 4 T + 18 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
31$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
37$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
41$D_{4}$ \( 1 - 4 T + 54 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 + 8 T + 70 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
47$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
53$D_{4}$ \( 1 + 16 T + 162 T^{2} + 16 p T^{3} + p^{2} T^{4} \)
59$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
67$D_{4}$ \( 1 + 8 T + 22 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 + 4 T + 74 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 - 4 T - 50 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
79$C_2^2$ \( 1 + 126 T^{2} + p^{2} T^{4} \)
83$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
89$C_4$ \( 1 + 12 T + 86 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 + 12 T + 222 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
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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

−11.11523555624672312666832883670, −10.97663637229730499427910546066, −10.36275811704857005484562393732, −9.829934421667500394177909520458, −9.588586046567756890086921749742, −9.154078141054155021048351273996, −8.488643266238240589053142416595, −8.337666645569010342128560282527, −7.56015941543733011576267120619, −7.40320449581073324832387392396, −6.86312831015611519146735414807, −6.12522467857803522401585965672, −5.41029474882458852985978086104, −4.96352485699437409756559981220, −4.08738209618549570146385865450, −3.55582162773919786837256595214, −2.71804868450455840630447744204, −1.94094429730931879137389705152, 0, 0, 1.94094429730931879137389705152, 2.71804868450455840630447744204, 3.55582162773919786837256595214, 4.08738209618549570146385865450, 4.96352485699437409756559981220, 5.41029474882458852985978086104, 6.12522467857803522401585965672, 6.86312831015611519146735414807, 7.40320449581073324832387392396, 7.56015941543733011576267120619, 8.337666645569010342128560282527, 8.488643266238240589053142416595, 9.154078141054155021048351273996, 9.588586046567756890086921749742, 9.829934421667500394177909520458, 10.36275811704857005484562393732, 10.97663637229730499427910546066, 11.11523555624672312666832883670

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