Properties

Label 4-2850e2-1.1-c1e2-0-13
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

Downloads

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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 + 20·41-s − 8·44-s − 2·49-s − 8·59-s − 20·61-s − 64-s − 32·71-s + 2·76-s − 20·79-s + 81-s + 4·89-s − 8·99-s + 16·101-s + 8·109-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 + 3.12·41-s − 1.20·44-s − 2/7·49-s − 1.04·59-s − 2.56·61-s − 1/8·64-s − 3.79·71-s + 0.229·76-s − 2.25·79-s + 1/9·81-s + 0.423·89-s − 0.804·99-s + 1.59·101-s + 0.766·109-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

\(L(1)\) \(\approx\) \(2.826713003\)
\(L(\frac12)\) \(\approx\) \(2.826713003\)
\(L(\frac{3}{2})\) 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$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - p T^{2} )^{2} \)
17$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
23$C_2^2$ \( 1 - 42 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
37$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 + 58 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 102 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - p T^{2} )^{2} \)
71$C_2$ \( ( 1 + 16 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 142 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 + 90 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
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

−9.172678692651458236724416013991, −8.513403679622155291112088585860, −8.499159188035330344554895026198, −7.76252151051415262120386576000, −7.60799564017764026109824089240, −7.02454506213722211429463104947, −6.57048264420955247487545925404, −6.28061167244187554116487598269, −5.89364909250228128511606023411, −5.88314331476774635472977651492, −4.79139275687943335467170495903, −4.57608245003361116656499426239, −4.23663863485613693660125827639, −4.14161021826269028708360943105, −3.10796775056040567510113070050, −3.07568648604847498666448083071, −2.49789346200610138262890858836, −1.54305139657801473633269937797, −1.23801903575987554825865432874, −0.61976720010593364723207997820, 0.61976720010593364723207997820, 1.23801903575987554825865432874, 1.54305139657801473633269937797, 2.49789346200610138262890858836, 3.07568648604847498666448083071, 3.10796775056040567510113070050, 4.14161021826269028708360943105, 4.23663863485613693660125827639, 4.57608245003361116656499426239, 4.79139275687943335467170495903, 5.88314331476774635472977651492, 5.89364909250228128511606023411, 6.28061167244187554116487598269, 6.57048264420955247487545925404, 7.02454506213722211429463104947, 7.60799564017764026109824089240, 7.76252151051415262120386576000, 8.499159188035330344554895026198, 8.513403679622155291112088585860, 9.172678692651458236724416013991

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