Properties

Label 4-2450e2-1.1-c1e2-0-1
Degree $4$
Conductor $6002500$
Sign $1$
Analytic cond. $382.724$
Root an. cond. $4.42304$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 4-s + 2·9-s − 8·11-s + 16-s − 12·19-s + 12·29-s − 8·31-s − 2·36-s − 8·41-s + 8·44-s − 28·59-s + 20·61-s − 64-s + 24·71-s + 12·76-s − 8·79-s − 5·81-s + 16·89-s − 16·99-s + 4·101-s − 20·109-s − 12·116-s + 26·121-s + 8·124-s + 127-s + 131-s + 137-s + ⋯
L(s)  = 1  − 1/2·4-s + 2/3·9-s − 2.41·11-s + 1/4·16-s − 2.75·19-s + 2.22·29-s − 1.43·31-s − 1/3·36-s − 1.24·41-s + 1.20·44-s − 3.64·59-s + 2.56·61-s − 1/8·64-s + 2.84·71-s + 1.37·76-s − 0.900·79-s − 5/9·81-s + 1.69·89-s − 1.60·99-s + 0.398·101-s − 1.91·109-s − 1.11·116-s + 2.36·121-s + 0.718·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6002500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6002500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(6002500\)    =    \(2^{2} \cdot 5^{4} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(382.724\)
Root analytic conductor: \(4.42304\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 6002500,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2641808511\)
\(L(\frac12)\) \(\approx\) \(0.2641808511\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + T^{2} \)
5 \( 1 \)
7 \( 1 \)
good3$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
11$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
13$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
17$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
19$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
37$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 78 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 - 118 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 162 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - p T^{2} )^{2} \)
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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

−9.014619622579347709460126090766, −8.728233298460355061167773961378, −8.178980632104023768692069085125, −8.153518513887687755753002447629, −7.75542122949507699534015337561, −7.29555338852647573165789085839, −6.67651918697707293120022713533, −6.52700820152537373034893095128, −6.12211488576219172631154649251, −5.43778116358633702680744602733, −5.04778809685292755127277083359, −4.96629606893722603634935763330, −4.27112550085034187750845063547, −4.11271534635929546705211120193, −3.38891401383221457119677660786, −2.92118403382524317181678753742, −2.24411964701753057802777684427, −2.14706211275372369188052938703, −1.21782147781835079564142420615, −0.17757336305304888764839935047, 0.17757336305304888764839935047, 1.21782147781835079564142420615, 2.14706211275372369188052938703, 2.24411964701753057802777684427, 2.92118403382524317181678753742, 3.38891401383221457119677660786, 4.11271534635929546705211120193, 4.27112550085034187750845063547, 4.96629606893722603634935763330, 5.04778809685292755127277083359, 5.43778116358633702680744602733, 6.12211488576219172631154649251, 6.52700820152537373034893095128, 6.67651918697707293120022713533, 7.29555338852647573165789085839, 7.75542122949507699534015337561, 8.153518513887687755753002447629, 8.178980632104023768692069085125, 8.728233298460355061167773961378, 9.014619622579347709460126090766

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