Properties

Label 4-880e2-1.1-c1e2-0-30
Degree $4$
Conductor $774400$
Sign $1$
Analytic cond. $49.3764$
Root an. cond. $2.65081$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·5-s + 2·9-s − 2·11-s + 8·19-s − 25-s + 4·29-s − 16·31-s + 12·41-s + 4·45-s − 2·49-s − 4·55-s − 8·59-s − 20·61-s + 16·71-s + 8·79-s − 5·81-s + 28·89-s + 16·95-s − 4·99-s + 20·101-s + 28·109-s + 3·121-s − 12·125-s + ⋯
L(s)  = 1  + 0.894·5-s + 2/3·9-s − 0.603·11-s + 1.83·19-s − 1/5·25-s + 0.742·29-s − 2.87·31-s + 1.87·41-s + 0.596·45-s − 2/7·49-s − 0.539·55-s − 1.04·59-s − 2.56·61-s + 1.89·71-s + 0.900·79-s − 5/9·81-s + 2.96·89-s + 1.64·95-s − 0.402·99-s + 1.99·101-s + 2.68·109-s + 3/11·121-s − 1.07·125-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 774400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 774400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(774400\)    =    \(2^{8} \cdot 5^{2} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(49.3764\)
Root analytic conductor: \(2.65081\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 774400,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.494781883\)
\(L(\frac12)\) \(\approx\) \(2.494781883\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5$C_2$ \( 1 - 2 T + p T^{2} \)
11$C_1$ \( ( 1 + T )^{2} \)
good3$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.3.a_ac
7$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.7.a_c
13$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.13.a_k
17$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.17.a_abe
19$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.19.ai_cc
23$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.23.a_ak
29$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.29.ae_ck
31$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.31.q_ew
37$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.37.a_ak
41$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.41.am_eo
43$C_2^2$ \( 1 + 58 T^{2} + p^{2} T^{4} \) 2.43.a_cg
47$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \) 2.47.a_g
53$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.53.a_aec
59$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.59.i_fe
61$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.61.u_io
67$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \) 2.67.a_afa
71$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.71.aq_hy
73$C_2^2$ \( 1 - 142 T^{2} + p^{2} T^{4} \) 2.73.a_afm
79$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.79.ai_gs
83$C_2^2$ \( 1 - 150 T^{2} + p^{2} T^{4} \) 2.83.a_afu
89$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \) 2.89.abc_ok
97$C_2^2$ \( 1 - 178 T^{2} + p^{2} T^{4} \) 2.97.a_agw
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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

−10.45803462893824144279072245183, −9.693964136396894298368878541269, −9.573130176757961113313804112957, −9.262215190844250844017593468252, −8.876367017848573481397020461682, −8.166757413117946499717744869165, −7.57063300568067126383469153718, −7.48005622337731277326750103828, −7.14917821214173835529832113229, −6.22752483856707988059263737864, −6.08342573587508510184956483734, −5.59469975944140291401404749680, −4.92032197604678558133236977117, −4.87544534309813936394287031326, −3.96807518765375229986214574327, −3.42194278218698302765357204583, −2.96986766000025597926792122965, −2.10531266647646563688591560793, −1.72783014168056431748981072373, −0.78412738756074621289558862968, 0.78412738756074621289558862968, 1.72783014168056431748981072373, 2.10531266647646563688591560793, 2.96986766000025597926792122965, 3.42194278218698302765357204583, 3.96807518765375229986214574327, 4.87544534309813936394287031326, 4.92032197604678558133236977117, 5.59469975944140291401404749680, 6.08342573587508510184956483734, 6.22752483856707988059263737864, 7.14917821214173835529832113229, 7.48005622337731277326750103828, 7.57063300568067126383469153718, 8.166757413117946499717744869165, 8.876367017848573481397020461682, 9.262215190844250844017593468252, 9.573130176757961113313804112957, 9.693964136396894298368878541269, 10.45803462893824144279072245183

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