Properties

Label 4-2850e2-1.1-c1e2-0-9
Degree $4$
Conductor $8122500$
Sign $1$
Analytic cond. $517.897$
Root an. cond. $4.77046$
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 − 9-s + 16-s − 2·19-s − 12·29-s + 4·31-s + 36-s + 12·41-s − 2·49-s + 24·59-s + 28·61-s − 64-s + 2·76-s + 20·79-s + 81-s + 12·89-s + 32·109-s + 12·116-s − 22·121-s − 4·124-s + 127-s + 131-s + 137-s + 139-s − 144-s + 149-s + 151-s + ⋯
L(s)  = 1  − 1/2·4-s − 1/3·9-s + 1/4·16-s − 0.458·19-s − 2.22·29-s + 0.718·31-s + 1/6·36-s + 1.87·41-s − 2/7·49-s + 3.12·59-s + 3.58·61-s − 1/8·64-s + 0.229·76-s + 2.25·79-s + 1/9·81-s + 1.27·89-s + 3.06·109-s + 1.11·116-s − 2·121-s − 0.359·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 0.0833·144-s + 0.0819·149-s + 0.0813·151-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8122500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8122500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(8122500\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{4} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(517.897\)
Root analytic conductor: \(4.77046\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{2850} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 8122500,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.904614134\)
\(L(\frac12)\) \(\approx\) \(1.904614134\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + T^{2} \)
3$C_2$ \( 1 + T^{2} \)
5 \( 1 \)
19$C_1$ \( ( 1 + T )^{2} \)
good7$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
11$C_2$ \( ( 1 + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
17$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
23$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
37$C_2^2$ \( 1 - 58 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 58 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2^2$ \( 1 + 50 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
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

−8.762601137633688930135565974823, −8.707924546066177169490290949439, −8.320189834051550421184466012893, −7.84790887721582945729195550272, −7.45388836060171032454165352073, −7.21488981173211428289124788735, −6.69157433898784711523042244955, −6.13319976906138436544438566732, −6.06219415243581732653210482748, −5.36276075021088364313858603462, −5.15144791684935465233929567914, −4.83656939869977604559892487213, −4.01701106225434364186231577418, −3.74572846576651904593028088118, −3.71737891447654124832833402850, −2.75440449176235264017235152792, −2.29966579180626510198145790575, −2.01538314346297996997146711218, −1.00450375836979626562020635740, −0.53151822188718079537270376493, 0.53151822188718079537270376493, 1.00450375836979626562020635740, 2.01538314346297996997146711218, 2.29966579180626510198145790575, 2.75440449176235264017235152792, 3.71737891447654124832833402850, 3.74572846576651904593028088118, 4.01701106225434364186231577418, 4.83656939869977604559892487213, 5.15144791684935465233929567914, 5.36276075021088364313858603462, 6.06219415243581732653210482748, 6.13319976906138436544438566732, 6.69157433898784711523042244955, 7.21488981173211428289124788735, 7.45388836060171032454165352073, 7.84790887721582945729195550272, 8.320189834051550421184466012893, 8.707924546066177169490290949439, 8.762601137633688930135565974823

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