Properties

Label 4-5070e2-1.1-c1e2-0-1
Degree $4$
Conductor $25704900$
Sign $1$
Analytic cond. $1638.96$
Root an. cond. $6.36271$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s − 4-s + 3·9-s + 2·12-s + 16-s + 8·23-s − 25-s − 4·27-s − 3·36-s + 4·43-s − 2·48-s + 5·49-s − 26·53-s − 4·61-s − 64-s − 16·69-s + 2·75-s − 20·79-s + 5·81-s − 8·92-s + 100-s − 8·101-s + 18·103-s − 12·107-s + 4·108-s + 32·113-s + 21·121-s + ⋯
L(s)  = 1  − 1.15·3-s − 1/2·4-s + 9-s + 0.577·12-s + 1/4·16-s + 1.66·23-s − 1/5·25-s − 0.769·27-s − 1/2·36-s + 0.609·43-s − 0.288·48-s + 5/7·49-s − 3.57·53-s − 0.512·61-s − 1/8·64-s − 1.92·69-s + 0.230·75-s − 2.25·79-s + 5/9·81-s − 0.834·92-s + 1/10·100-s − 0.796·101-s + 1.77·103-s − 1.16·107-s + 0.384·108-s + 3.01·113-s + 1.90·121-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 25704900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25704900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(25704900\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 13^{4}\)
Sign: $1$
Analytic conductor: \(1638.96\)
Root analytic conductor: \(6.36271\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{5070} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 25704900,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5233194886\)
\(L(\frac12)\) \(\approx\) \(0.5233194886\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 + T )^{2} \)
5$C_2$ \( 1 + T^{2} \)
13 \( 1 \)
good7$C_2^2$ \( 1 - 5 T^{2} + p^{2} T^{4} \)
11$C_2^2$ \( 1 - 21 T^{2} + p^{2} T^{4} \)
17$C_2$ \( ( 1 + p T^{2} )^{2} \)
19$C_2^2$ \( 1 - 13 T^{2} + p^{2} T^{4} \)
23$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
29$C_2$ \( ( 1 + p T^{2} )^{2} \)
31$C_2^2$ \( 1 + 38 T^{2} + p^{2} T^{4} \)
37$C_2^2$ \( 1 - 73 T^{2} + p^{2} T^{4} \)
41$C_2^2$ \( 1 - 46 T^{2} + p^{2} T^{4} \)
43$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 - 13 T^{2} + p^{2} T^{4} \)
53$C_2$ \( ( 1 + 13 T + p T^{2} )^{2} \)
59$C_2^2$ \( 1 - 102 T^{2} + p^{2} T^{4} \)
61$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \)
71$C_2^2$ \( 1 - 138 T^{2} + p^{2} T^{4} \)
73$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
79$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
89$C_2^2$ \( 1 - 177 T^{2} + p^{2} T^{4} \)
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

−8.624455572400189252113725503015, −7.890366130486890043551756101203, −7.67582679953680771177977052102, −7.28116974565876885695440782314, −6.99872038361984360056429601323, −6.51242010608576055860759163538, −6.22694060213884888219550318361, −5.85568131055190993830714123800, −5.54170786465685025551120874055, −5.09009880763016517780845986568, −4.64830485734619457871118172859, −4.61578999069602632734149920491, −4.10920311006230030776778162191, −3.38394702273183657784340783046, −3.31062367643283639200029230308, −2.64017374191209443633296919649, −2.08655138042056272995658128936, −1.26677341228616871085555583614, −1.18608190003359032364789492095, −0.24066229365829699213152901968, 0.24066229365829699213152901968, 1.18608190003359032364789492095, 1.26677341228616871085555583614, 2.08655138042056272995658128936, 2.64017374191209443633296919649, 3.31062367643283639200029230308, 3.38394702273183657784340783046, 4.10920311006230030776778162191, 4.61578999069602632734149920491, 4.64830485734619457871118172859, 5.09009880763016517780845986568, 5.54170786465685025551120874055, 5.85568131055190993830714123800, 6.22694060213884888219550318361, 6.51242010608576055860759163538, 6.99872038361984360056429601323, 7.28116974565876885695440782314, 7.67582679953680771177977052102, 7.890366130486890043551756101203, 8.624455572400189252113725503015

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