Properties

Label 4-12e6-1.1-c0e2-0-0
Degree $4$
Conductor $2985984$
Sign $1$
Analytic cond. $0.743706$
Root an. cond. $0.928646$
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  − 2·13-s − 2·25-s + 2·37-s − 49-s + 2·61-s + 2·73-s + 2·97-s + 4·109-s + 2·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯
L(s)  = 1  − 2·13-s − 2·25-s + 2·37-s − 49-s + 2·61-s + 2·73-s + 2·97-s + 4·109-s + 2·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2985984 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2985984 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(2985984\)    =    \(2^{12} \cdot 3^{6}\)
Sign: $1$
Analytic conductor: \(0.743706\)
Root analytic conductor: \(0.928646\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1728} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 2985984,\ (\ :0, 0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8983210171\)
\(L(\frac12)\) \(\approx\) \(0.8983210171\)
\(L(1)\) 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 \( 1 \)
3 \( 1 \)
good5$C_2$ \( ( 1 + T^{2} )^{2} \)
7$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
11$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
13$C_2$ \( ( 1 + T + T^{2} )^{2} \)
17$C_2$ \( ( 1 + T^{2} )^{2} \)
19$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
23$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
29$C_2$ \( ( 1 + T^{2} )^{2} \)
31$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
37$C_2$ \( ( 1 - T + T^{2} )^{2} \)
41$C_2$ \( ( 1 + T^{2} )^{2} \)
43$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
47$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
53$C_2$ \( ( 1 + T^{2} )^{2} \)
59$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
61$C_2$ \( ( 1 - T + T^{2} )^{2} \)
67$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
71$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
73$C_2$ \( ( 1 - T + T^{2} )^{2} \)
79$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
83$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
89$C_2$ \( ( 1 + T^{2} )^{2} \)
97$C_2$ \( ( 1 - T + T^{2} )^{2} \)
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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

−9.701759286508287861707756639335, −9.615462340917795172134851838264, −8.933819712347643075619844316162, −8.443455406363999126001194511581, −8.159215719417198043085060233607, −7.50737926119342817179520179736, −7.49363298184073124659858536979, −7.14788637360418958420857473214, −6.26784205467010861434491546022, −6.26522248990561709667051410290, −5.72506125508415380321200066592, −4.98734926646215308428529286052, −4.96311573602473579778484235358, −4.39354425764661822369255022095, −3.80139067778985993802534900300, −3.45412206338884521975917905470, −2.65943885781688702367402266851, −2.26899311024382299705108684468, −1.88351757240761996605787485158, −0.71002701286702548198936457465, 0.71002701286702548198936457465, 1.88351757240761996605787485158, 2.26899311024382299705108684468, 2.65943885781688702367402266851, 3.45412206338884521975917905470, 3.80139067778985993802534900300, 4.39354425764661822369255022095, 4.96311573602473579778484235358, 4.98734926646215308428529286052, 5.72506125508415380321200066592, 6.26522248990561709667051410290, 6.26784205467010861434491546022, 7.14788637360418958420857473214, 7.49363298184073124659858536979, 7.50737926119342817179520179736, 8.159215719417198043085060233607, 8.443455406363999126001194511581, 8.933819712347643075619844316162, 9.615462340917795172134851838264, 9.701759286508287861707756639335

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