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Properties

Label 4-5175e2-1.1-c1e2-0-11
Degree $4$
Conductor $26780625$
Sign $1$
Analytic cond. $1707.55$
Root an. cond. $6.42826$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + 4-s + 4·7-s − 8·14-s + 16-s − 12·17-s − 4·19-s − 2·23-s + 4·28-s + 2·32-s + 24·34-s + 4·37-s + 8·38-s + 8·41-s + 12·43-s + 4·46-s + 12·47-s − 4·53-s + 4·59-s + 4·61-s − 11·64-s − 20·67-s − 12·68-s − 16·71-s − 4·73-s − 8·74-s − 4·76-s + ⋯
L(s)  = 1  − 1.41·2-s + 1/2·4-s + 1.51·7-s − 2.13·14-s + 1/4·16-s − 2.91·17-s − 0.917·19-s − 0.417·23-s + 0.755·28-s + 0.353·32-s + 4.11·34-s + 0.657·37-s + 1.29·38-s + 1.24·41-s + 1.82·43-s + 0.589·46-s + 1.75·47-s − 0.549·53-s + 0.520·59-s + 0.512·61-s − 1.37·64-s − 2.44·67-s − 1.45·68-s − 1.89·71-s − 0.468·73-s − 0.929·74-s − 0.458·76-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 26780625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 26780625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(26780625\)    =    \(3^{4} \cdot 5^{4} \cdot 23^{2}\)
Sign: $1$
Analytic conductor: \(1707.55\)
Root analytic conductor: \(6.42826\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 26780625,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 $\mathbf{F}_p$
bad3 \( 1 \)
5 \( 1 \)
23$C_1$ \( ( 1 + T )^{2} \)
good2$D_{4}$ \( 1 + p T + 3 T^{2} + p^{2} T^{3} + p^{2} T^{4} \) 2.2.c_d
7$D_{4}$ \( 1 - 4 T + 16 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.7.ae_q
11$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \) 2.11.a_o
13$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.13.a_ba
17$D_{4}$ \( 1 + 12 T + 4 p T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.17.m_cq
19$D_{4}$ \( 1 + 4 T + 24 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.19.e_y
29$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \) 2.29.a_ao
31$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.31.a_ak
37$D_{4}$ \( 1 - 4 T + 70 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.37.ae_cs
41$D_{4}$ \( 1 - 8 T + 66 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.41.ai_co
43$D_{4}$ \( 1 - 12 T + 104 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.43.am_ea
47$D_{4}$ \( 1 - 12 T + 98 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.47.am_du
53$D_{4}$ \( 1 + 4 T + 60 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.53.e_ci
59$D_{4}$ \( 1 - 4 T + 90 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.59.ae_dm
61$D_{4}$ \( 1 - 4 T + 118 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.61.ae_eo
67$D_{4}$ \( 1 + 20 T + 232 T^{2} + 20 p T^{3} + p^{2} T^{4} \) 2.67.u_iy
71$D_{4}$ \( 1 + 16 T + 174 T^{2} + 16 p T^{3} + p^{2} T^{4} \) 2.71.q_gs
73$D_{4}$ \( 1 + 4 T + 22 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.73.e_w
79$D_{4}$ \( 1 + 4 T + 64 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.79.e_cm
83$D_{4}$ \( 1 + 8 T + 174 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.83.i_gs
89$D_{4}$ \( 1 - 12 T + 164 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.89.am_gi
97$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \) 2.97.au_li
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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

−8.117403323211770402038231137101, −7.73914699908265542027964603347, −7.44212969276821585006702069987, −7.29387035625081921525449015230, −6.64437448599803901801705901421, −6.17879355020552293570222627654, −6.06218994199601855879909151256, −5.69411785641999205567842101980, −4.82095177899972167728205822734, −4.71959930178482882515599064814, −4.41159938244136802471650755422, −4.09171494343731411724442051418, −3.60761456529587723116679668347, −2.60747539863987050435024064441, −2.41719498694984989674356133806, −2.21744297483320697043169794135, −1.30494872758807112445736760302, −1.17652089248666232955865544009, 0, 0, 1.17652089248666232955865544009, 1.30494872758807112445736760302, 2.21744297483320697043169794135, 2.41719498694984989674356133806, 2.60747539863987050435024064441, 3.60761456529587723116679668347, 4.09171494343731411724442051418, 4.41159938244136802471650755422, 4.71959930178482882515599064814, 4.82095177899972167728205822734, 5.69411785641999205567842101980, 6.06218994199601855879909151256, 6.17879355020552293570222627654, 6.64437448599803901801705901421, 7.29387035625081921525449015230, 7.44212969276821585006702069987, 7.73914699908265542027964603347, 8.117403323211770402038231137101

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