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Properties

Label 4-5376e2-1.1-c1e2-0-17
Degree $4$
Conductor $28901376$
Sign $1$
Analytic cond. $1842.77$
Root an. cond. $6.55191$
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  + 2·7-s − 9-s + 12·23-s + 6·25-s + 8·41-s − 16·47-s + 3·49-s − 2·63-s + 12·71-s + 20·73-s + 81-s − 28·97-s − 16·103-s − 12·113-s + 22·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 24·161-s + 163-s + 167-s + 26·169-s + 173-s + ⋯
L(s)  = 1  + 0.755·7-s − 1/3·9-s + 2.50·23-s + 6/5·25-s + 1.24·41-s − 2.33·47-s + 3/7·49-s − 0.251·63-s + 1.42·71-s + 2.34·73-s + 1/9·81-s − 2.84·97-s − 1.57·103-s − 1.12·113-s + 2·121-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 + 1.89·161-s + 0.0783·163-s + 0.0773·167-s + 2·169-s + 0.0760·173-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 28901376 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28901376 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(28901376\)    =    \(2^{16} \cdot 3^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(1842.77\)
Root analytic conductor: \(6.55191\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 28901376,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.206103813\)
\(L(\frac12)\) \(\approx\) \(3.206103813\)
\(L(\frac{3}{2})\) 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)$Isogeny Class over $\mathbf{F}_p$
bad2 \( 1 \)
3$C_2$ \( 1 + T^{2} \)
7$C_1$ \( ( 1 - T )^{2} \)
good5$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.5.a_ag
11$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.11.a_aw
13$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.13.a_aba
17$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.17.a_bi
19$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \) 2.19.a_aw
23$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.23.am_de
29$C_2^2$ \( 1 - 54 T^{2} + p^{2} T^{4} \) 2.29.a_acc
31$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.31.a_ck
37$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.37.a_acs
41$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.41.ai_du
43$C_2^2$ \( 1 - 82 T^{2} + p^{2} T^{4} \) 2.43.a_ade
47$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.47.q_gc
53$C_2^2$ \( 1 - 102 T^{2} + p^{2} T^{4} \) 2.53.a_ady
59$C_2^2$ \( 1 - 102 T^{2} + p^{2} T^{4} \) 2.59.a_ady
61$C_2^2$ \( 1 - 58 T^{2} + p^{2} T^{4} \) 2.61.a_acg
67$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \) 2.67.a_abi
71$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.71.am_gw
73$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \) 2.73.au_jm
79$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.79.a_gc
83$C_2^2$ \( 1 - 150 T^{2} + p^{2} T^{4} \) 2.83.a_afu
89$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.89.a_gw
97$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \) 2.97.bc_pa
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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

−8.395741912001371956819628310906, −8.000353381878619036790957916009, −7.78439896490761952128379440382, −7.16696083748123591290599086818, −6.93937441504709718051577003506, −6.62211916352716495129799527957, −6.38759074151353713177549970233, −5.64741630789959615869042758384, −5.45898724049720995372233139718, −4.99749876201679971901622117659, −4.84345097603506664326813463747, −4.42207316066328367752068191857, −3.91714185194938434222110712223, −3.28838660108542139215064469015, −3.19779946530125404744260478560, −2.45176894348710131164619845479, −2.36934680621902340384130475777, −1.37030158295408208278010045528, −1.22962484513991110304231013265, −0.51388270376163982729963428991, 0.51388270376163982729963428991, 1.22962484513991110304231013265, 1.37030158295408208278010045528, 2.36934680621902340384130475777, 2.45176894348710131164619845479, 3.19779946530125404744260478560, 3.28838660108542139215064469015, 3.91714185194938434222110712223, 4.42207316066328367752068191857, 4.84345097603506664326813463747, 4.99749876201679971901622117659, 5.45898724049720995372233139718, 5.64741630789959615869042758384, 6.38759074151353713177549970233, 6.62211916352716495129799527957, 6.93937441504709718051577003506, 7.16696083748123591290599086818, 7.78439896490761952128379440382, 8.000353381878619036790957916009, 8.395741912001371956819628310906

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