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Properties

Label 4-42e4-1.1-c1e2-0-20
Degree $4$
Conductor $3111696$
Sign $1$
Analytic cond. $198.404$
Root an. cond. $3.75308$
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  − 3·3-s − 6·5-s + 6·9-s + 6·11-s − 13-s + 18·15-s + 6·17-s − 4·19-s − 6·23-s + 17·25-s − 9·27-s − 3·29-s + 5·31-s − 18·33-s − 2·37-s + 3·39-s + 3·41-s + 43-s − 36·45-s − 9·47-s − 18·51-s + 6·53-s − 36·55-s + 12·57-s − 3·59-s − 13·61-s + 6·65-s + ⋯
L(s)  = 1  − 1.73·3-s − 2.68·5-s + 2·9-s + 1.80·11-s − 0.277·13-s + 4.64·15-s + 1.45·17-s − 0.917·19-s − 1.25·23-s + 17/5·25-s − 1.73·27-s − 0.557·29-s + 0.898·31-s − 3.13·33-s − 0.328·37-s + 0.480·39-s + 0.468·41-s + 0.152·43-s − 5.36·45-s − 1.31·47-s − 2.52·51-s + 0.824·53-s − 4.85·55-s + 1.58·57-s − 0.390·59-s − 1.66·61-s + 0.744·65-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3111696 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3111696 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(3111696\)    =    \(2^{4} \cdot 3^{4} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(198.404\)
Root analytic conductor: \(3.75308\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 3111696,\ (\ :1/2, 1/2),\ 1)\)

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$C_2$ \( 1 + p T + p T^{2} \)
7 \( 1 \)
good5$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \) 2.5.g_t
11$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \) 2.11.ag_bf
13$C_2^2$ \( 1 + T - 12 T^{2} + p T^{3} + p^{2} T^{4} \) 2.13.b_am
17$C_2^2$ \( 1 - 6 T + 19 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.17.ag_t
19$C_2^2$ \( 1 + 4 T - 3 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.19.e_ad
23$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \) 2.23.g_cd
29$C_2^2$ \( 1 + 3 T - 20 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.29.d_au
31$C_2^2$ \( 1 - 5 T - 6 T^{2} - 5 p T^{3} + p^{2} T^{4} \) 2.31.af_ag
37$C_2^2$ \( 1 + 2 T - 33 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.37.c_abh
41$C_2^2$ \( 1 - 3 T - 32 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.41.ad_abg
43$C_2^2$ \( 1 - T - 42 T^{2} - p T^{3} + p^{2} T^{4} \) 2.43.ab_abq
47$C_2^2$ \( 1 + 9 T + 34 T^{2} + 9 p T^{3} + p^{2} T^{4} \) 2.47.j_bi
53$C_2^2$ \( 1 - 6 T - 17 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.53.ag_ar
59$C_2^2$ \( 1 + 3 T - 50 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.59.d_aby
61$C_2$ \( ( 1 - T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.61.n_ee
67$C_2^2$ \( 1 - 7 T - 18 T^{2} - 7 p T^{3} + p^{2} T^{4} \) 2.67.ah_as
71$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \) 2.71.y_la
73$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 17 T + p T^{2} ) \) 2.73.k_bb
79$C_2^2$ \( 1 + 11 T + 42 T^{2} + 11 p T^{3} + p^{2} T^{4} \) 2.79.l_bq
83$C_2^2$ \( 1 + 9 T - 2 T^{2} + 9 p T^{3} + p^{2} T^{4} \) 2.83.j_ac
89$C_2^2$ \( 1 - 6 T - 53 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.89.ag_acb
97$C_2^2$ \( 1 - 11 T + 24 T^{2} - 11 p T^{3} + p^{2} T^{4} \) 2.97.al_y
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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

−8.959763530829391837183297999517, −8.710206504266210235463390436096, −8.117051941293410790627888932049, −7.82014685789549243214842514264, −7.46808525755896647109077909280, −7.21044560837577064408029856653, −6.61339915518368447320487477451, −6.40918126890972494567761537091, −5.85029034320608533759182380018, −5.59320563302622552579647413090, −4.68518777546796800208285697480, −4.61451726972303206318621844708, −3.99649753670924468970330636455, −3.93881560402874991141100323373, −3.45356563246760492536841584847, −2.73340419430328274153309556371, −1.47150915248539676189471536097, −1.19679183310770055757348002303, 0, 0, 1.19679183310770055757348002303, 1.47150915248539676189471536097, 2.73340419430328274153309556371, 3.45356563246760492536841584847, 3.93881560402874991141100323373, 3.99649753670924468970330636455, 4.61451726972303206318621844708, 4.68518777546796800208285697480, 5.59320563302622552579647413090, 5.85029034320608533759182380018, 6.40918126890972494567761537091, 6.61339915518368447320487477451, 7.21044560837577064408029856653, 7.46808525755896647109077909280, 7.82014685789549243214842514264, 8.117051941293410790627888932049, 8.710206504266210235463390436096, 8.959763530829391837183297999517

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