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Properties

Label 4-2376e2-1.1-c1e2-0-5
Degree $4$
Conductor $5645376$
Sign $1$
Analytic cond. $359.954$
Root an. cond. $4.35573$
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  − 2·7-s − 2·11-s − 8·13-s − 2·17-s + 4·19-s + 6·23-s − 2·25-s + 10·29-s − 4·31-s − 2·37-s + 6·41-s + 18·43-s + 10·47-s − 9·49-s + 8·53-s + 10·59-s + 12·67-s + 16·71-s − 16·73-s + 4·77-s − 2·79-s − 8·83-s − 4·89-s + 16·91-s − 6·97-s + 14·101-s + 16·107-s + ⋯
L(s)  = 1  − 0.755·7-s − 0.603·11-s − 2.21·13-s − 0.485·17-s + 0.917·19-s + 1.25·23-s − 2/5·25-s + 1.85·29-s − 0.718·31-s − 0.328·37-s + 0.937·41-s + 2.74·43-s + 1.45·47-s − 9/7·49-s + 1.09·53-s + 1.30·59-s + 1.46·67-s + 1.89·71-s − 1.87·73-s + 0.455·77-s − 0.225·79-s − 0.878·83-s − 0.423·89-s + 1.67·91-s − 0.609·97-s + 1.39·101-s + 1.54·107-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5645376 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5645376 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(5645376\)    =    \(2^{6} \cdot 3^{6} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(359.954\)
Root analytic conductor: \(4.35573\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 5645376,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.654192732\)
\(L(\frac12)\) \(\approx\) \(1.654192732\)
\(L(\frac{3}{2})\) 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 \( 1 \)
11$C_1$ \( ( 1 + T )^{2} \)
good5$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.5.a_c
7$D_{4}$ \( 1 + 2 T + 13 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.7.c_n
13$D_{4}$ \( 1 + 8 T + 34 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.13.i_bi
17$D_{4}$ \( 1 + 2 T + p T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.17.c_r
19$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.19.ae_bq
23$D_{4}$ \( 1 - 6 T + p T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.23.ag_x
29$D_{4}$ \( 1 - 10 T + 81 T^{2} - 10 p T^{3} + p^{2} T^{4} \) 2.29.ak_dd
31$D_{4}$ \( 1 + 4 T + 34 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.31.e_bi
37$D_{4}$ \( 1 + 2 T + 43 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.37.c_br
41$D_{4}$ \( 1 - 6 T + 89 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.41.ag_dl
43$D_{4}$ \( 1 - 18 T + 165 T^{2} - 18 p T^{3} + p^{2} T^{4} \) 2.43.as_gj
47$D_{4}$ \( 1 - 10 T + p T^{2} - 10 p T^{3} + p^{2} T^{4} \) 2.47.ak_bv
53$D_{4}$ \( 1 - 8 T + 90 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.53.ai_dm
59$D_{4}$ \( 1 - 10 T + 135 T^{2} - 10 p T^{3} + p^{2} T^{4} \) 2.59.ak_ff
61$C_2^2$ \( 1 + 114 T^{2} + p^{2} T^{4} \) 2.61.a_ek
67$D_{4}$ \( 1 - 12 T + 138 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.67.am_fi
71$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.71.aq_hy
73$D_{4}$ \( 1 + 16 T + 178 T^{2} + 16 p T^{3} + p^{2} T^{4} \) 2.73.q_gw
79$D_{4}$ \( 1 + 2 T + 157 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.79.c_gb
83$D_{4}$ \( 1 + 8 T + 174 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.83.i_gs
89$D_{4}$ \( 1 + 4 T + 110 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.89.e_eg
97$D_{4}$ \( 1 + 6 T + 3 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.97.g_d
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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

−9.194413723693101817136337691698, −8.962397062716025990859092357269, −8.314180901703231061596393422644, −8.049236057585858808282002789627, −7.36929618316651139669927332519, −7.31648370869902037313656460888, −6.99463645187778340985572052491, −6.61774441469176549158155663394, −5.84090314487282502000679274273, −5.76644242261418359068531854558, −5.06965802686053415309056386090, −4.97811878239635449426036863335, −4.33994127992736739568310987032, −4.03455350085871187487666451502, −3.27096964856999486497907290274, −2.88513358102742298353679852342, −2.37844633849722411842774557871, −2.27051431439103808775659467086, −1.02423192564320377147991882936, −0.51047271635026393279328848180, 0.51047271635026393279328848180, 1.02423192564320377147991882936, 2.27051431439103808775659467086, 2.37844633849722411842774557871, 2.88513358102742298353679852342, 3.27096964856999486497907290274, 4.03455350085871187487666451502, 4.33994127992736739568310987032, 4.97811878239635449426036863335, 5.06965802686053415309056386090, 5.76644242261418359068531854558, 5.84090314487282502000679274273, 6.61774441469176549158155663394, 6.99463645187778340985572052491, 7.31648370869902037313656460888, 7.36929618316651139669927332519, 8.049236057585858808282002789627, 8.314180901703231061596393422644, 8.962397062716025990859092357269, 9.194413723693101817136337691698

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