Properties

Degree $4$
Conductor $49787136$
Sign $1$
Motivic weight $1$
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 12·11-s + 8·23-s − 2·25-s + 12·29-s + 4·37-s − 20·43-s + 4·53-s − 8·67-s − 24·71-s + 8·79-s + 8·107-s − 20·109-s − 8·113-s + 86·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 6·169-s + 173-s + 179-s + 181-s + ⋯
L(s)  = 1  + 3.61·11-s + 1.66·23-s − 2/5·25-s + 2.22·29-s + 0.657·37-s − 3.04·43-s + 0.549·53-s − 0.977·67-s − 2.84·71-s + 0.900·79-s + 0.773·107-s − 1.91·109-s − 0.752·113-s + 7.81·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 6/13·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 49787136 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49787136 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(49787136\)    =    \(2^{8} \cdot 3^{4} \cdot 7^{4}\)
Sign: $1$
Motivic weight: \(1\)
Character: induced by $\chi_{7056} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 49787136,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.952790294\)
\(L(\frac12)\) \(\approx\) \(4.952790294\)
\(L(\frac{3}{2})\) 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 \( 1 \)
3 \( 1 \)
7 \( 1 \)
good5$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
11$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
13$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \)
17$C_2^2$ \( 1 + 32 T^{2} + p^{2} T^{4} \)
19$C_2^2$ \( 1 + 20 T^{2} + p^{2} T^{4} \)
23$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
31$C_2^2$ \( 1 + 54 T^{2} + p^{2} T^{4} \)
37$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 + 80 T^{2} + p^{2} T^{4} \)
43$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 + 86 T^{2} + p^{2} T^{4} \)
53$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
59$C_2^2$ \( 1 + 116 T^{2} + p^{2} T^{4} \)
61$C_2^2$ \( 1 + 50 T^{2} + p^{2} T^{4} \)
67$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 + 48 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 + 164 T^{2} + p^{2} T^{4} \)
89$C_2^2$ \( 1 + 160 T^{2} + p^{2} T^{4} \)
97$C_2^2$ \( 1 + 32 T^{2} + p^{2} T^{4} \)
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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

−8.186606903107144253527389982554, −7.82632224636268707738844512862, −7.17671109456718182259662692951, −6.86449818490525449453570564035, −6.83599943364003155639979165473, −6.48807244167066257063252337390, −6.04388864948741487810013134854, −5.89996236354136227367083182381, −5.22714853778186210804765791226, −4.75681735409154423647814216043, −4.50341937202362248346770819380, −4.27212020799128708050598395436, −3.62953030193807424433779539844, −3.52902903902973866602280651938, −2.96865577631073130815384270789, −2.66961043669717636699923991565, −1.69263622372052197459451285730, −1.57661125791959474201251567119, −1.11981705315670534212033177055, −0.61227819689763455025954233183, 0.61227819689763455025954233183, 1.11981705315670534212033177055, 1.57661125791959474201251567119, 1.69263622372052197459451285730, 2.66961043669717636699923991565, 2.96865577631073130815384270789, 3.52902903902973866602280651938, 3.62953030193807424433779539844, 4.27212020799128708050598395436, 4.50341937202362248346770819380, 4.75681735409154423647814216043, 5.22714853778186210804765791226, 5.89996236354136227367083182381, 6.04388864948741487810013134854, 6.48807244167066257063252337390, 6.83599943364003155639979165473, 6.86449818490525449453570564035, 7.17671109456718182259662692951, 7.82632224636268707738844512862, 8.186606903107144253527389982554

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