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Properties

Label 4-3520e2-1.1-c0e2-0-7
Degree $4$
Conductor $12390400$
Sign $1$
Analytic cond. $3.08602$
Root an. cond. $1.32540$
Motivic weight $0$
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  + 2·3-s + 2·9-s − 2·23-s − 25-s + 2·27-s + 2·37-s + 2·47-s + 2·53-s − 4·59-s − 2·67-s − 4·69-s − 2·75-s + 3·81-s + 2·97-s + 2·103-s + 4·111-s + 2·113-s − 121-s + 127-s + 131-s + 137-s + 139-s + 4·141-s + 149-s + 151-s + 157-s + 4·159-s + ⋯
L(s)  = 1  + 2·3-s + 2·9-s − 2·23-s − 25-s + 2·27-s + 2·37-s + 2·47-s + 2·53-s − 4·59-s − 2·67-s − 4·69-s − 2·75-s + 3·81-s + 2·97-s + 2·103-s + 4·111-s + 2·113-s − 121-s + 127-s + 131-s + 137-s + 139-s + 4·141-s + 149-s + 151-s + 157-s + 4·159-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 12390400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12390400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(12390400\)    =    \(2^{12} \cdot 5^{2} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(3.08602\)
Root analytic conductor: \(1.32540\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 12390400,\ (\ :0, 0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.897207898\)
\(L(\frac12)\) \(\approx\) \(2.897207898\)
\(L(1)\) 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$C_2$ \( 1 + T^{2} \)
11$C_2$ \( 1 + T^{2} \)
good3$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T^{2} ) \)
7$C_2^2$ \( 1 + T^{4} \)
13$C_2^2$ \( 1 + T^{4} \)
17$C_2^2$ \( 1 + T^{4} \)
19$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
23$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 + T^{2} ) \)
29$C_2$ \( ( 1 + T^{2} )^{2} \)
31$C_2$ \( ( 1 + T^{2} )^{2} \)
37$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T^{2} ) \)
41$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
43$C_2^2$ \( 1 + T^{4} \)
47$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T^{2} ) \)
53$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T^{2} ) \)
59$C_1$ \( ( 1 + T )^{4} \)
61$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
67$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 + T^{2} ) \)
71$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
73$C_2^2$ \( 1 + T^{4} \)
79$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
83$C_2^2$ \( 1 + T^{4} \)
89$C_2$ \( ( 1 + T^{2} )^{2} \)
97$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T^{2} ) \)
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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

−8.959413830882432028919420140206, −8.426804945928555706539348257693, −8.373121607941518872698173430142, −7.71306184159860454152181016566, −7.57432548371909978402923575997, −7.51366561642598946347535082897, −6.87904312531357524354101443112, −6.12002223052367527014788395765, −5.98997272316989416539037392683, −5.89491416418119443838896801987, −4.93672759219392977097245953046, −4.52294321957145122556567156863, −4.26871364490257042912855529142, −3.75576818628344640257844640158, −3.49148663243138970128139299962, −2.75972577348477865730905137322, −2.74235442698613735742205764445, −1.95986582775280986915833989711, −1.87878290293833525195146886545, −0.884116654868285036603463869265, 0.884116654868285036603463869265, 1.87878290293833525195146886545, 1.95986582775280986915833989711, 2.74235442698613735742205764445, 2.75972577348477865730905137322, 3.49148663243138970128139299962, 3.75576818628344640257844640158, 4.26871364490257042912855529142, 4.52294321957145122556567156863, 4.93672759219392977097245953046, 5.89491416418119443838896801987, 5.98997272316989416539037392683, 6.12002223052367527014788395765, 6.87904312531357524354101443112, 7.51366561642598946347535082897, 7.57432548371909978402923575997, 7.71306184159860454152181016566, 8.373121607941518872698173430142, 8.426804945928555706539348257693, 8.959413830882432028919420140206

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