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Properties

Label 4-6e8-1.1-c1e2-0-39
Degree $4$
Conductor $1679616$
Sign $1$
Analytic cond. $107.093$
Root an. cond. $3.21692$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·7-s − 2·13-s − 4·19-s − 7·25-s − 16·31-s − 14·37-s − 4·43-s − 2·49-s − 14·61-s + 20·67-s − 14·73-s − 4·79-s + 8·91-s + 4·97-s − 16·103-s + 22·109-s − 10·121-s + 127-s + 131-s + 16·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯
L(s)  = 1  − 1.51·7-s − 0.554·13-s − 0.917·19-s − 7/5·25-s − 2.87·31-s − 2.30·37-s − 0.609·43-s − 2/7·49-s − 1.79·61-s + 2.44·67-s − 1.63·73-s − 0.450·79-s + 0.838·91-s + 0.406·97-s − 1.57·103-s + 2.10·109-s − 0.909·121-s + 0.0887·127-s + 0.0873·131-s + 1.38·133-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1679616 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1679616 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1679616\)    =    \(2^{8} \cdot 3^{8}\)
Sign: $1$
Analytic conductor: \(107.093\)
Root analytic conductor: \(3.21692\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 1679616,\ (\ :1/2, 1/2),\ 1)\)

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 \( 1 \)
3 \( 1 \)
good5$C_2^2$ \( 1 + 7 T^{2} + p^{2} T^{4} \) 2.5.a_h
7$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.7.e_s
11$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \) 2.11.a_k
13$C_2$ \( ( 1 + T + p T^{2} )^{2} \) 2.13.c_bb
17$C_2^2$ \( 1 + 7 T^{2} + p^{2} T^{4} \) 2.17.a_h
19$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.19.e_bq
23$C_2^2$ \( 1 + 34 T^{2} + p^{2} T^{4} \) 2.23.a_bi
29$C_2^2$ \( 1 + 55 T^{2} + p^{2} T^{4} \) 2.29.a_cd
31$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.31.q_ew
37$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \) 2.37.o_et
41$C_2^2$ \( 1 + 34 T^{2} + p^{2} T^{4} \) 2.41.a_bi
43$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.43.e_dm
47$C_2^2$ \( 1 + 46 T^{2} + p^{2} T^{4} \) 2.47.a_bu
53$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.53.a_ec
59$C_2^2$ \( 1 - 74 T^{2} + p^{2} T^{4} \) 2.59.a_acw
61$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \) 2.61.o_gp
67$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \) 2.67.au_ja
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$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.79.e_gg
83$C_2^2$ \( 1 - 26 T^{2} + p^{2} T^{4} \) 2.83.a_aba
89$C_2^2$ \( 1 + 151 T^{2} + p^{2} T^{4} \) 2.89.a_fv
97$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.97.ae_hq
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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

−9.341665840237041808285026809603, −9.251543305674722385981349876713, −8.653035129377198706582112656228, −8.389618765664295620574504065647, −7.63268012183850984179346547688, −7.47705274032406590905081998415, −6.78852207522708156626236004309, −6.78270554031334006495476278895, −6.03698972320889941748641237324, −5.85920264864410306753396467976, −5.17040795997156123977108790290, −4.94777782237655644122297906753, −4.02781749984666884065129793051, −3.78266019077985299822337629759, −3.33095289294352973300171301447, −2.81804376389584014178196517575, −1.97260392726912140594628395778, −1.69982390901808710962200993785, 0, 0, 1.69982390901808710962200993785, 1.97260392726912140594628395778, 2.81804376389584014178196517575, 3.33095289294352973300171301447, 3.78266019077985299822337629759, 4.02781749984666884065129793051, 4.94777782237655644122297906753, 5.17040795997156123977108790290, 5.85920264864410306753396467976, 6.03698972320889941748641237324, 6.78270554031334006495476278895, 6.78852207522708156626236004309, 7.47705274032406590905081998415, 7.63268012183850984179346547688, 8.389618765664295620574504065647, 8.653035129377198706582112656228, 9.251543305674722385981349876713, 9.341665840237041808285026809603

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