Properties

Label 4-2850e2-1.1-c1e2-0-8
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 + 8·11-s + 16-s − 2·19-s − 12·29-s − 12·31-s + 36-s + 8·41-s − 8·44-s + 10·49-s − 20·59-s + 4·61-s − 64-s + 16·71-s + 2·76-s − 20·79-s + 81-s + 16·89-s − 8·99-s − 20·101-s + 32·109-s + 12·116-s + 26·121-s + 12·124-s + 127-s + 131-s + ⋯
L(s)  = 1  − 1/2·4-s − 1/3·9-s + 2.41·11-s + 1/4·16-s − 0.458·19-s − 2.22·29-s − 2.15·31-s + 1/6·36-s + 1.24·41-s − 1.20·44-s + 10/7·49-s − 2.60·59-s + 0.512·61-s − 1/8·64-s + 1.89·71-s + 0.229·76-s − 2.25·79-s + 1/9·81-s + 1.69·89-s − 0.804·99-s − 1.99·101-s + 3.06·109-s + 1.11·116-s + 2.36·121-s + 1.07·124-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

\(L(1)\) \(\approx\) \(1.724117511\)
\(L(\frac12)\) \(\approx\) \(1.724117511\)
\(L(\frac{3}{2})\) 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$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
17$C_2^2$ \( 1 - 18 T^{2} + p^{2} T^{4} \)
23$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
37$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 + 58 T^{2} + p^{2} T^{4} \)
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$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 142 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 162 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
97$C_2^2$ \( 1 - 190 T^{2} + p^{2} T^{4} \)
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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

−9.076232949796361748004452299009, −8.589688826068793192431022972043, −8.526674192127450870900585074983, −7.67625203087583607427196638117, −7.36618445495933633407945798707, −7.30971579522236546171354458437, −6.62464738580503012037546813772, −6.29415300425127812129712328931, −5.83313231248563889231148090436, −5.68957911853166994285362254139, −5.11788191287211178584690861608, −4.58478537367037455842870717051, −4.08514646910782990938764816347, −3.84801924277571135919746412430, −3.57984029163808962078857393061, −3.00743367109339990223384674192, −2.17338521756582501508382990039, −1.78706012495652740781046567398, −1.26968061586183733942077707688, −0.44587884870362194364063710397, 0.44587884870362194364063710397, 1.26968061586183733942077707688, 1.78706012495652740781046567398, 2.17338521756582501508382990039, 3.00743367109339990223384674192, 3.57984029163808962078857393061, 3.84801924277571135919746412430, 4.08514646910782990938764816347, 4.58478537367037455842870717051, 5.11788191287211178584690861608, 5.68957911853166994285362254139, 5.83313231248563889231148090436, 6.29415300425127812129712328931, 6.62464738580503012037546813772, 7.30971579522236546171354458437, 7.36618445495933633407945798707, 7.67625203087583607427196638117, 8.526674192127450870900585074983, 8.589688826068793192431022972043, 9.076232949796361748004452299009

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