Properties

Degree $4$
Conductor $12960000$
Sign $1$
Motivic weight $0$
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 + 4·37-s − 49-s − 2·61-s − 4·73-s + 2·97-s + 2·109-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 + 223-s + ⋯
L(s)  = 1  + 2·13-s + 4·37-s − 49-s − 2·61-s − 4·73-s + 2·97-s + 2·109-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 + 223-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 12960000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12960000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(12960000\)    =    \(2^{8} \cdot 3^{4} \cdot 5^{4}\)
Sign: $1$
Motivic weight: \(0\)
Character: induced by $\chi_{3600} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 12960000,\ (\ :0, 0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.701829136\)
\(L(\frac12)\) \(\approx\) \(1.701829136\)
\(L(1)\) 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 \( 1 \)
5 \( 1 \)
good7$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
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$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
37$C_1$ \( ( 1 - T )^{4} \)
41$C_2$ \( ( 1 + T^{2} )^{2} \)
43$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
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$C_1$ \( ( 1 + T )^{4} \)
79$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
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

−8.867470330904173342821232056490, −8.446156141950173316095183383653, −8.352937788875759211196280498756, −7.74150735439921101569748939819, −7.39502693616187491454632631477, −7.34431156178424980269602863901, −6.41932832409070900917269557885, −6.25529837166572875014060175006, −5.96532382474648780660299390712, −5.85717804278547754481236016725, −5.10156698455657904980448646174, −4.53293702398748791021384047300, −4.41151213172758678057851222304, −3.96765265705382171201653111116, −3.35147616920509415702730398561, −3.06362229102243986605066833457, −2.61659039085290545781143792099, −1.88976216955315927353710890275, −1.36858935655896603797884666969, −0.849287628200568591356291987110, 0.849287628200568591356291987110, 1.36858935655896603797884666969, 1.88976216955315927353710890275, 2.61659039085290545781143792099, 3.06362229102243986605066833457, 3.35147616920509415702730398561, 3.96765265705382171201653111116, 4.41151213172758678057851222304, 4.53293702398748791021384047300, 5.10156698455657904980448646174, 5.85717804278547754481236016725, 5.96532382474648780660299390712, 6.25529837166572875014060175006, 6.41932832409070900917269557885, 7.34431156178424980269602863901, 7.39502693616187491454632631477, 7.74150735439921101569748939819, 8.352937788875759211196280498756, 8.446156141950173316095183383653, 8.867470330904173342821232056490

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