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Properties

Label 4-2366e2-1.1-c1e2-0-6
Degree $4$
Conductor $5597956$
Sign $1$
Analytic cond. $356.930$
Root an. cond. $4.34656$
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  − 4·3-s − 4-s + 6·9-s + 4·12-s + 16-s + 6·17-s + 12·23-s + 25-s + 4·27-s + 6·29-s − 6·36-s + 20·43-s − 4·48-s − 49-s − 24·51-s − 18·53-s − 14·61-s − 64-s − 6·68-s − 48·69-s − 4·75-s + 28·79-s − 37·81-s − 24·87-s − 12·92-s − 100-s + 6·101-s + ⋯
L(s)  = 1  − 2.30·3-s − 1/2·4-s + 2·9-s + 1.15·12-s + 1/4·16-s + 1.45·17-s + 2.50·23-s + 1/5·25-s + 0.769·27-s + 1.11·29-s − 36-s + 3.04·43-s − 0.577·48-s − 1/7·49-s − 3.36·51-s − 2.47·53-s − 1.79·61-s − 1/8·64-s − 0.727·68-s − 5.77·69-s − 0.461·75-s + 3.15·79-s − 4.11·81-s − 2.57·87-s − 1.25·92-s − 0.0999·100-s + 0.597·101-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5597956 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5597956 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(5597956\)    =    \(2^{2} \cdot 7^{2} \cdot 13^{4}\)
Sign: $1$
Analytic conductor: \(356.930\)
Root analytic conductor: \(4.34656\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 5597956,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8743580536\)
\(L(\frac12)\) \(\approx\) \(0.8743580536\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + T^{2} \)
7$C_2$ \( 1 + T^{2} \)
13 \( 1 \)
good3$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.3.e_k
5$C_2^2$ \( 1 - T^{2} + p^{2} T^{4} \) 2.5.a_ab
11$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \) 2.11.a_o
17$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \) 2.17.ag_br
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$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \) 2.29.ag_cp
31$C_2^2$ \( 1 + 38 T^{2} + p^{2} T^{4} \) 2.31.a_bm
37$C_2^2$ \( 1 - 73 T^{2} + p^{2} T^{4} \) 2.37.a_acv
41$C_2^2$ \( 1 - 73 T^{2} + p^{2} T^{4} \) 2.41.a_acv
43$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \) 2.43.au_he
47$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.47.a_adq
53$C_2$ \( ( 1 + 9 T + p T^{2} )^{2} \) 2.53.s_hf
59$C_2^2$ \( 1 - 82 T^{2} + p^{2} T^{4} \) 2.59.a_ade
61$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \) 2.61.o_gp
67$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \) 2.67.a_afa
71$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.71.a_afm
73$C_2^2$ \( 1 - 97 T^{2} + p^{2} T^{4} \) 2.73.a_adt
79$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \) 2.79.abc_nq
83$C_2^2$ \( 1 + 158 T^{2} + p^{2} T^{4} \) 2.83.a_gc
89$C_2^2$ \( 1 - 142 T^{2} + p^{2} T^{4} \) 2.89.a_afm
97$C_2^2$ \( 1 - 190 T^{2} + p^{2} T^{4} \) 2.97.a_ahi
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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

−9.176456279280782499421801297033, −9.021815247235888260409217854704, −8.386287621983200504108223070574, −7.87155755998979876522036464228, −7.61381376898637989668792239146, −7.17024844502121251948035204392, −6.63263329455942970948921197321, −6.25243398642438979155868653800, −6.13206886124576596910222488987, −5.61729243399358276735797768547, −5.15352417927156515130287855948, −4.92766449933949695691335550189, −4.76421043167002611112525172731, −4.13325425901026073184629392674, −3.44727137957389734558285298878, −2.96661305666930403912590126360, −2.61360966318602513556049747684, −1.35758904984638082493168359807, −0.988347163041432410860143448174, −0.51367760153837726363666324369, 0.51367760153837726363666324369, 0.988347163041432410860143448174, 1.35758904984638082493168359807, 2.61360966318602513556049747684, 2.96661305666930403912590126360, 3.44727137957389734558285298878, 4.13325425901026073184629392674, 4.76421043167002611112525172731, 4.92766449933949695691335550189, 5.15352417927156515130287855948, 5.61729243399358276735797768547, 6.13206886124576596910222488987, 6.25243398642438979155868653800, 6.63263329455942970948921197321, 7.17024844502121251948035204392, 7.61381376898637989668792239146, 7.87155755998979876522036464228, 8.386287621983200504108223070574, 9.021815247235888260409217854704, 9.176456279280782499421801297033

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