Properties

Label 4-80e4-1.1-c1e2-0-27
Degree $4$
Conductor $40960000$
Sign $1$
Analytic cond. $2611.64$
Root an. cond. $7.14872$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s − 3·9-s − 6·11-s + 6·17-s + 2·19-s + 14·27-s + 12·33-s + 18·41-s − 8·43-s − 2·49-s − 12·51-s − 4·57-s + 24·59-s − 22·67-s − 14·73-s − 4·81-s − 30·83-s + 6·89-s + 28·97-s + 18·99-s − 18·107-s − 30·113-s + 5·121-s − 36·123-s + ⋯
L(s)  = 1  − 1.15·3-s − 9-s − 1.80·11-s + 1.45·17-s + 0.458·19-s + 2.69·27-s + 2.08·33-s + 2.81·41-s − 1.21·43-s − 2/7·49-s − 1.68·51-s − 0.529·57-s + 3.12·59-s − 2.68·67-s − 1.63·73-s − 4/9·81-s − 3.29·83-s + 0.635·89-s + 2.84·97-s + 1.80·99-s − 1.74·107-s − 2.82·113-s + 5/11·121-s − 3.24·123-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 40960000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40960000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(40960000\)    =    \(2^{16} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(2611.64\)
Root analytic conductor: \(7.14872\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 40960000,\ (\ :1/2, 1/2),\ 1)\)

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 $\mathbf{F}_p$
bad2 \( 1 \)
5 \( 1 \)
good3$C_2$ \( ( 1 + T + p T^{2} )^{2} \) 2.3.c_h
7$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.7.a_c
11$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \) 2.11.g_bf
13$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \) 2.13.a_o
17$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \) 2.17.ag_br
19$C_2$ \( ( 1 - T + p T^{2} )^{2} \) 2.19.ac_bn
23$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.23.a_bu
29$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \) 2.29.a_aby
31$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \) 2.31.a_o
37$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \) 2.37.a_abi
41$C_2$ \( ( 1 - 9 T + p T^{2} )^{2} \) 2.41.as_gh
43$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.43.i_dy
47$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \) 2.47.a_ao
53$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.53.a_ec
59$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \) 2.59.ay_kc
61$C_2^2$ \( 1 + 110 T^{2} + p^{2} T^{4} \) 2.61.a_eg
67$C_2$ \( ( 1 + 11 T + p T^{2} )^{2} \) 2.67.w_jv
71$C_2^2$ \( 1 + 34 T^{2} + p^{2} T^{4} \) 2.71.a_bi
73$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \) 2.73.o_hn
79$C_2^2$ \( 1 + 50 T^{2} + p^{2} T^{4} \) 2.79.a_by
83$C_2$ \( ( 1 + 15 T + p T^{2} )^{2} \) 2.83.be_pb
89$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \) 2.89.ag_hf
97$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \) 2.97.abc_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

−7.78780378961370343832532905981, −7.59730464086360264715983843332, −7.12801620229096879882879342161, −6.83242231925202286569576968901, −6.17662238146902434575253380162, −5.98413333545002015693438473382, −5.62920368284516816425504237223, −5.52626339177379673315659638160, −5.04318839647285545666921392296, −4.90624736158526718878273568931, −4.29540563998387000260642623835, −3.86729398559344607618242963697, −3.18710741915481272596207289518, −3.02128450589521995989985113229, −2.52033648665040371102930659032, −2.34186024121928306578464351097, −1.23166933464039151195080706374, −1.02851184903277241635924384639, 0, 0, 1.02851184903277241635924384639, 1.23166933464039151195080706374, 2.34186024121928306578464351097, 2.52033648665040371102930659032, 3.02128450589521995989985113229, 3.18710741915481272596207289518, 3.86729398559344607618242963697, 4.29540563998387000260642623835, 4.90624736158526718878273568931, 5.04318839647285545666921392296, 5.52626339177379673315659638160, 5.62920368284516816425504237223, 5.98413333545002015693438473382, 6.17662238146902434575253380162, 6.83242231925202286569576968901, 7.12801620229096879882879342161, 7.59730464086360264715983843332, 7.78780378961370343832532905981

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