Properties

Label 4-1900e2-1.1-c1e2-0-4
Degree $4$
Conductor $3610000$
Sign $1$
Analytic cond. $230.176$
Root an. cond. $3.89507$
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 + 8·7-s + 3·9-s − 6·11-s + 6·13-s + 2·17-s + 7·19-s + 16·21-s + 4·23-s + 10·27-s − 29-s − 10·31-s − 12·33-s + 8·37-s + 12·39-s − 2·41-s + 6·47-s + 34·49-s + 4·51-s − 6·53-s + 14·57-s + 59-s + 7·61-s + 24·63-s − 14·67-s + 8·69-s − 15·71-s + ⋯
L(s)  = 1  + 1.15·3-s + 3.02·7-s + 9-s − 1.80·11-s + 1.66·13-s + 0.485·17-s + 1.60·19-s + 3.49·21-s + 0.834·23-s + 1.92·27-s − 0.185·29-s − 1.79·31-s − 2.08·33-s + 1.31·37-s + 1.92·39-s − 0.312·41-s + 0.875·47-s + 34/7·49-s + 0.560·51-s − 0.824·53-s + 1.85·57-s + 0.130·59-s + 0.896·61-s + 3.02·63-s − 1.71·67-s + 0.963·69-s − 1.78·71-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3610000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3610000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(3610000\)    =    \(2^{4} \cdot 5^{4} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(230.176\)
Root analytic conductor: \(3.89507\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1900} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 3610000,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(7.083881262\)
\(L(\frac12)\) \(\approx\) \(7.083881262\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 \)
19$C_2$ \( 1 - 7 T + p T^{2} \)
good3$C_2^2$ \( 1 - 2 T + T^{2} - 2 p T^{3} + p^{2} T^{4} \)
7$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
11$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
13$C_2^2$ \( 1 - 6 T + 23 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
17$C_2^2$ \( 1 - 2 T - 13 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
23$C_2^2$ \( 1 - 4 T - 7 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
29$C_2^2$ \( 1 + T - 28 T^{2} + p T^{3} + p^{2} T^{4} \)
31$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 + 2 T - 37 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
43$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 6 T - 11 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
53$C_2^2$ \( 1 + 6 T - 17 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 - T - 58 T^{2} - p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 - 7 T - 12 T^{2} - 7 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 + 14 T + 129 T^{2} + 14 p T^{3} + p^{2} T^{4} \)
71$C_2^2$ \( 1 + 15 T + 154 T^{2} + 15 p T^{3} + p^{2} T^{4} \)
73$C_2^2$ \( 1 - 12 T + 71 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
79$C_2^2$ \( 1 - T - 78 T^{2} - p T^{3} + p^{2} T^{4} \)
83$C_2$ \( ( 1 + 16 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 + 17 T + 200 T^{2} + 17 p T^{3} + p^{2} T^{4} \)
97$C_2^2$ \( 1 - 12 T + 47 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
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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

−9.185966347256608838486731586542, −8.885271539108395495634580380864, −8.403553996277668332390904360092, −8.262525055877581624746629679982, −8.028011200385701199778456805967, −7.54812121296920486526209381431, −7.20077439042336357277017919332, −7.14660415200485460840162573864, −5.98977008679118641934874384442, −5.69030909251564658771053480452, −5.16408250055960883457152531609, −5.11702550620692209216151035385, −4.35293500856012483902849609656, −4.26836777191941391460572853982, −3.40930429422636265560328362352, −3.02160327713235391472926172150, −2.55875957768212249296486296401, −1.88163309739496665316701301808, −1.35554462950371377971177577490, −1.08844407421819833641327426099, 1.08844407421819833641327426099, 1.35554462950371377971177577490, 1.88163309739496665316701301808, 2.55875957768212249296486296401, 3.02160327713235391472926172150, 3.40930429422636265560328362352, 4.26836777191941391460572853982, 4.35293500856012483902849609656, 5.11702550620692209216151035385, 5.16408250055960883457152531609, 5.69030909251564658771053480452, 5.98977008679118641934874384442, 7.14660415200485460840162573864, 7.20077439042336357277017919332, 7.54812121296920486526209381431, 8.028011200385701199778456805967, 8.262525055877581624746629679982, 8.403553996277668332390904360092, 8.885271539108395495634580380864, 9.185966347256608838486731586542

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