Properties

Label 4-1040e2-1.1-c1e2-0-19
Degree $4$
Conductor $1081600$
Sign $1$
Analytic cond. $68.9637$
Root an. cond. $2.88174$
Motivic weight $1$
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  + 2·5-s + 2·9-s − 6·13-s − 25-s + 12·29-s − 12·37-s + 4·45-s − 16·47-s − 14·49-s + 12·61-s − 12·65-s + 24·67-s + 12·73-s − 5·81-s − 8·83-s + 12·97-s + 12·101-s − 12·117-s + 18·121-s − 12·125-s + 127-s + 131-s + 137-s + 139-s + 24·145-s + 149-s + 151-s + ⋯
L(s)  = 1  + 0.894·5-s + 2/3·9-s − 1.66·13-s − 1/5·25-s + 2.22·29-s − 1.97·37-s + 0.596·45-s − 2.33·47-s − 2·49-s + 1.53·61-s − 1.48·65-s + 2.93·67-s + 1.40·73-s − 5/9·81-s − 0.878·83-s + 1.21·97-s + 1.19·101-s − 1.10·117-s + 1.63·121-s − 1.07·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 1.99·145-s + 0.0819·149-s + 0.0813·151-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1081600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1081600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1081600\)    =    \(2^{8} \cdot 5^{2} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(68.9637\)
Root analytic conductor: \(2.88174\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1040} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1081600,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.144180845\)
\(L(\frac12)\) \(\approx\) \(2.144180845\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - 2 T + p T^{2} \)
13$C_2$ \( 1 + 6 T + p T^{2} \)
good3$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
7$C_2$ \( ( 1 + p T^{2} )^{2} \)
11$C_2^2$ \( 1 - 18 T^{2} + p^{2} T^{4} \)
17$C_2$ \( ( 1 - p T^{2} )^{2} \)
19$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
23$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
31$C_2^2$ \( 1 - 26 T^{2} + p^{2} T^{4} \)
37$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
43$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \)
47$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
53$C_2^2$ \( 1 + 38 T^{2} + p^{2} T^{4} \)
59$C_2^2$ \( 1 - 114 T^{2} + p^{2} T^{4} \)
61$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
71$C_2^2$ \( 1 - 138 T^{2} + p^{2} T^{4} \)
73$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 + p T^{2} )^{2} \)
83$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 - 114 T^{2} + p^{2} T^{4} \)
97$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
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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

−10.04550523434964221350431791893, −9.792937377881083293575322033069, −9.628517546118631800794722278491, −8.859463484345468218341805707242, −8.512001745118887951581839009846, −8.024899303462589330315294464003, −7.75558050714700163873008419714, −6.87765604520897280122427321441, −6.86546055063972996429447570295, −6.52208621393245152975290275549, −5.87885084718917339429083425466, −5.10774852424646912255425059314, −5.08857877220442595065199326266, −4.65102999398781231804203433239, −3.94275227180622519613045851516, −3.26951814974367224974595473256, −2.82677117847101712530097713933, −1.94897118320689200813379395203, −1.84779343068204927971982666045, −0.65121989537002453233270200583, 0.65121989537002453233270200583, 1.84779343068204927971982666045, 1.94897118320689200813379395203, 2.82677117847101712530097713933, 3.26951814974367224974595473256, 3.94275227180622519613045851516, 4.65102999398781231804203433239, 5.08857877220442595065199326266, 5.10774852424646912255425059314, 5.87885084718917339429083425466, 6.52208621393245152975290275549, 6.86546055063972996429447570295, 6.87765604520897280122427321441, 7.75558050714700163873008419714, 8.024899303462589330315294464003, 8.512001745118887951581839009846, 8.859463484345468218341805707242, 9.628517546118631800794722278491, 9.792937377881083293575322033069, 10.04550523434964221350431791893

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