Properties

Label 4-2450e2-1.1-c1e2-0-9
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 + 6·9-s + 8·11-s + 16-s − 12·29-s − 16·31-s − 6·36-s − 4·41-s − 8·44-s − 16·59-s + 28·61-s − 64-s − 32·71-s + 16·79-s + 27·81-s + 20·89-s + 48·99-s + 12·101-s − 12·109-s + 12·116-s + 26·121-s + 16·124-s + 127-s + 131-s + 137-s + 139-s + 6·144-s + ⋯
L(s)  = 1  − 1/2·4-s + 2·9-s + 2.41·11-s + 1/4·16-s − 2.22·29-s − 2.87·31-s − 36-s − 0.624·41-s − 1.20·44-s − 2.08·59-s + 3.58·61-s − 1/8·64-s − 3.79·71-s + 1.80·79-s + 3·81-s + 2.11·89-s + 4.82·99-s + 1.19·101-s − 1.14·109-s + 1.11·116-s + 2.36·121-s + 1.43·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 1/2·144-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: 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

\(L(1)\) \(\approx\) \(2.865843463\)
\(L(\frac12)\) \(\approx\) \(2.865843463\)
\(L(\frac{3}{2})\) 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$C_2$ \( ( 1 - p T^{2} )^{2} \)
11$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
17$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
19$C_2$ \( ( 1 + p T^{2} )^{2} \)
23$C_2$ \( ( 1 - p T^{2} )^{2} \)
29$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
37$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 102 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + 16 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 142 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 102 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
97$C_2^2$ \( 1 - 190 T^{2} + p^{2} T^{4} \)
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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

−9.268353699802693065051488763483, −8.989726762343082665394647968083, −8.474702556584352627174212379876, −7.88699348270199839331785260553, −7.36346163818633471784470462890, −7.35723255013866401975578666122, −6.75455176798058391188704523467, −6.70621357434304784754136437266, −5.93812612562973689655658734544, −5.72049455523012331634926904893, −5.15400445526562305421510023393, −4.66023702078066563583429189332, −4.14732215425750944075815496081, −4.01466257195231029512624797866, −3.46279013586838455807860304301, −3.40127290978770289913697773548, −1.90579381612495718147824086067, −1.90173568409539422613977127687, −1.39990125129625124263851426966, −0.60619432205379762246235719692, 0.60619432205379762246235719692, 1.39990125129625124263851426966, 1.90173568409539422613977127687, 1.90579381612495718147824086067, 3.40127290978770289913697773548, 3.46279013586838455807860304301, 4.01466257195231029512624797866, 4.14732215425750944075815496081, 4.66023702078066563583429189332, 5.15400445526562305421510023393, 5.72049455523012331634926904893, 5.93812612562973689655658734544, 6.70621357434304784754136437266, 6.75455176798058391188704523467, 7.35723255013866401975578666122, 7.36346163818633471784470462890, 7.88699348270199839331785260553, 8.474702556584352627174212379876, 8.989726762343082665394647968083, 9.268353699802693065051488763483

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