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Properties

Label 4-588e2-1.1-c1e2-0-6
Degree $4$
Conductor $345744$
Sign $1$
Analytic cond. $22.0449$
Root an. cond. $2.16684$
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  + 2·2-s − 2·3-s + 2·4-s − 4·6-s + 3·9-s − 4·12-s − 4·16-s + 6·18-s − 2·19-s − 6·25-s − 4·27-s + 8·29-s − 18·31-s − 8·32-s + 6·36-s + 6·37-s − 4·38-s + 12·47-s + 8·48-s − 12·50-s + 24·53-s − 8·54-s + 4·57-s + 16·58-s + 24·59-s − 36·62-s − 8·64-s + ⋯
L(s)  = 1  + 1.41·2-s − 1.15·3-s + 4-s − 1.63·6-s + 9-s − 1.15·12-s − 16-s + 1.41·18-s − 0.458·19-s − 6/5·25-s − 0.769·27-s + 1.48·29-s − 3.23·31-s − 1.41·32-s + 36-s + 0.986·37-s − 0.648·38-s + 1.75·47-s + 1.15·48-s − 1.69·50-s + 3.29·53-s − 1.08·54-s + 0.529·57-s + 2.10·58-s + 3.12·59-s − 4.57·62-s − 64-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 345744 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 345744 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(345744\)    =    \(2^{4} \cdot 3^{2} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(22.0449\)
Root analytic conductor: \(2.16684\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 345744,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.090109359\)
\(L(\frac12)\) \(\approx\) \(2.090109359\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - p T + p T^{2} \)
3$C_1$ \( ( 1 + T )^{2} \)
7 \( 1 \)
good5$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.5.a_g
11$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.11.a_s
13$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) 2.13.a_z
17$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.17.a_bi
19$C_2$ \( ( 1 + T + p T^{2} )^{2} \) 2.19.c_bn
23$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.23.a_bu
29$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.29.ai_cw
31$C_2$ \( ( 1 + 9 T + p T^{2} )^{2} \) 2.31.s_fn
37$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \) 2.37.ag_df
41$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.41.a_as
43$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.43.a_cj
47$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.47.am_fa
53$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \) 2.53.ay_jq
59$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \) 2.59.ay_kc
61$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.61.a_w
67$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.67.a_ef
71$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.71.a_ec
73$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.73.a_fh
79$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) 2.79.a_gb
83$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.83.m_hu
89$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) 2.89.a_ada
97$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.97.a_gc
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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

−8.828648151345596153736554101594, −8.281244559015482682332319229048, −7.29485403540511127428179965078, −7.29254795769706133610428023090, −6.78325537260034591368785398088, −6.01621490934601452163404663718, −5.84278690300589592864666069852, −5.40855511243246313730427714163, −5.03090203649817702025598064487, −4.18739539622885934722496613081, −4.04530586363603743226969074324, −3.51834980769699570316939757660, −2.47062022887811000287520417928, −2.04117357649968623475261976339, −0.69023867370329802034851223268, 0.69023867370329802034851223268, 2.04117357649968623475261976339, 2.47062022887811000287520417928, 3.51834980769699570316939757660, 4.04530586363603743226969074324, 4.18739539622885934722496613081, 5.03090203649817702025598064487, 5.40855511243246313730427714163, 5.84278690300589592864666069852, 6.01621490934601452163404663718, 6.78325537260034591368785398088, 7.29254795769706133610428023090, 7.29485403540511127428179965078, 8.281244559015482682332319229048, 8.828648151345596153736554101594

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