Properties

Degree $4$
Conductor $6002500$
Sign $1$
Motivic weight $1$
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 4-s + 5·9-s + 6·11-s + 16-s − 14·19-s + 12·29-s + 8·31-s − 5·36-s + 18·41-s − 6·44-s + 24·59-s + 20·61-s − 64-s + 12·71-s + 14·76-s − 28·79-s + 16·81-s − 30·89-s + 30·99-s − 28·109-s − 12·116-s + 5·121-s − 8·124-s + 127-s + 131-s + 137-s + 139-s + ⋯
L(s)  = 1  − 1/2·4-s + 5/3·9-s + 1.80·11-s + 1/4·16-s − 3.21·19-s + 2.22·29-s + 1.43·31-s − 5/6·36-s + 2.81·41-s − 0.904·44-s + 3.12·59-s + 2.56·61-s − 1/8·64-s + 1.42·71-s + 1.60·76-s − 3.15·79-s + 16/9·81-s − 3.17·89-s + 3.01·99-s − 2.68·109-s − 1.11·116-s + 5/11·121-s − 0.718·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-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: $1$
Motivic weight: \(1\)
Character: induced by $\chi_{2450} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 6002500,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.300154721\)
\(L(\frac12)\) \(\approx\) \(3.300154721\)
\(L(\frac{3}{2})\) 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$C_2^2$ \( 1 - 5 T^{2} + p^{2} T^{4} \)
11$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
13$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
17$C_2^2$ \( 1 - 25 T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 - p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
37$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 - 9 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 58 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 + 38 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 - 85 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 121 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 85 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 + 15 T + p T^{2} )^{2} \)
97$C_2^2$ \( 1 - 94 T^{2} + p^{2} T^{4} \)
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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

−9.060403820093175901978861093502, −8.566767828534576884876706659597, −8.490532124872863079464037018826, −8.251577106685596022461795390188, −7.58151874902949358064537688703, −7.04842833833590880277776721134, −6.70162105190977187375468416688, −6.48129634868578868636556016924, −6.35922549668731555721178004746, −5.60306702001252148003430190571, −5.16460923396670724988183339617, −4.34178168622102925364911426728, −4.33478590734276780168099009521, −3.98041734576428427686310747813, −3.95806924599184681134781725135, −2.63760473033918254025977603676, −2.58014810226331224510076452612, −1.76616264277131175403557650458, −1.13931559641777263878283601826, −0.73743426868524249296873055799, 0.73743426868524249296873055799, 1.13931559641777263878283601826, 1.76616264277131175403557650458, 2.58014810226331224510076452612, 2.63760473033918254025977603676, 3.95806924599184681134781725135, 3.98041734576428427686310747813, 4.33478590734276780168099009521, 4.34178168622102925364911426728, 5.16460923396670724988183339617, 5.60306702001252148003430190571, 6.35922549668731555721178004746, 6.48129634868578868636556016924, 6.70162105190977187375468416688, 7.04842833833590880277776721134, 7.58151874902949358064537688703, 8.251577106685596022461795390188, 8.490532124872863079464037018826, 8.566767828534576884876706659597, 9.060403820093175901978861093502

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