Properties

Label 4-2850e2-1.1-c1e2-0-11
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 − 6·11-s + 16-s + 2·19-s + 10·29-s + 14·31-s + 36-s + 4·41-s + 6·44-s + 10·49-s + 20·59-s − 26·61-s − 64-s + 24·71-s − 2·76-s − 10·79-s + 81-s − 10·89-s + 6·99-s + 4·101-s − 20·109-s − 10·116-s + 5·121-s − 14·124-s + 127-s + 131-s + ⋯
L(s)  = 1  − 1/2·4-s − 1/3·9-s − 1.80·11-s + 1/4·16-s + 0.458·19-s + 1.85·29-s + 2.51·31-s + 1/6·36-s + 0.624·41-s + 0.904·44-s + 10/7·49-s + 2.60·59-s − 3.32·61-s − 1/8·64-s + 2.84·71-s − 0.229·76-s − 1.12·79-s + 1/9·81-s − 1.05·89-s + 0.603·99-s + 0.398·101-s − 1.91·109-s − 0.928·116-s + 5/11·121-s − 1.25·124-s + 0.0887·127-s + 0.0873·131-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

\(L(1)\) \(\approx\) \(1.958439271\)
\(L(\frac12)\) \(\approx\) \(1.958439271\)
\(L(\frac{3}{2})\) 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$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
11$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
17$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
23$C_2^2$ \( 1 - 45 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 7 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
41$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 + 50 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 25 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + 13 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 - 85 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 65 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 165 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \)
97$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \)
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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

−8.876953180679145020259005279157, −8.462279199523224918592703960594, −8.227442610722895579362172838116, −7.921242446631984372756002799708, −7.69332819003604671499366066574, −6.98396662738868629969933946053, −6.79702286144541205171218715456, −6.28027786695121415592994528889, −5.79098249384282533703610905507, −5.53247227159167075779657588176, −4.97362263765330483404157859474, −4.85462015337871508270560184677, −4.21588923682454297543402975749, −4.01830198106976228824132513125, −3.05589714247615099722976532279, −2.87140058729412711720948942795, −2.61166241734859765591218639953, −1.89372479066211228136237285980, −0.931711828038443377798866922169, −0.57507951696368228021810819184, 0.57507951696368228021810819184, 0.931711828038443377798866922169, 1.89372479066211228136237285980, 2.61166241734859765591218639953, 2.87140058729412711720948942795, 3.05589714247615099722976532279, 4.01830198106976228824132513125, 4.21588923682454297543402975749, 4.85462015337871508270560184677, 4.97362263765330483404157859474, 5.53247227159167075779657588176, 5.79098249384282533703610905507, 6.28027786695121415592994528889, 6.79702286144541205171218715456, 6.98396662738868629969933946053, 7.69332819003604671499366066574, 7.921242446631984372756002799708, 8.227442610722895579362172838116, 8.462279199523224918592703960594, 8.876953180679145020259005279157

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