Properties

Label 4-1386e2-1.1-c1e2-0-25
Degree $4$
Conductor $1920996$
Sign $1$
Analytic cond. $122.484$
Root an. cond. $3.32675$
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  + 2·2-s + 3·4-s − 2·7-s + 4·8-s − 2·11-s + 4·13-s − 4·14-s + 5·16-s + 8·17-s + 4·19-s − 4·22-s + 8·26-s − 6·28-s + 8·29-s + 12·31-s + 6·32-s + 16·34-s + 4·37-s + 8·38-s − 8·41-s − 6·44-s + 4·47-s + 3·49-s + 12·52-s − 4·53-s − 8·56-s + 16·58-s + ⋯
L(s)  = 1  + 1.41·2-s + 3/2·4-s − 0.755·7-s + 1.41·8-s − 0.603·11-s + 1.10·13-s − 1.06·14-s + 5/4·16-s + 1.94·17-s + 0.917·19-s − 0.852·22-s + 1.56·26-s − 1.13·28-s + 1.48·29-s + 2.15·31-s + 1.06·32-s + 2.74·34-s + 0.657·37-s + 1.29·38-s − 1.24·41-s − 0.904·44-s + 0.583·47-s + 3/7·49-s + 1.66·52-s − 0.549·53-s − 1.06·56-s + 2.10·58-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1920996 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1920996 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1920996\)    =    \(2^{2} \cdot 3^{4} \cdot 7^{2} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(122.484\)
Root analytic conductor: \(3.32675\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1920996,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(6.770945159\)
\(L(\frac12)\) \(\approx\) \(6.770945159\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 - T )^{2} \)
3 \( 1 \)
7$C_1$ \( ( 1 + T )^{2} \)
11$C_1$ \( ( 1 + T )^{2} \)
good5$C_2^2$ \( 1 + p^{2} T^{4} \) 2.5.a_a
13$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.13.ae_be
17$D_{4}$ \( 1 - 8 T + 40 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.17.ai_bo
19$D_{4}$ \( 1 - 4 T + 32 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.19.ae_bg
23$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \) 2.23.a_g
29$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.29.ai_cw
31$D_{4}$ \( 1 - 12 T + 88 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.31.am_dk
37$D_{4}$ \( 1 - 4 T + 38 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.37.ae_bm
41$D_{4}$ \( 1 + 8 T + 88 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.41.i_dk
43$C_2^2$ \( 1 + 46 T^{2} + p^{2} T^{4} \) 2.43.a_bu
47$C_2^2$ \( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.47.ae_i
53$D_{4}$ \( 1 + 4 T + 70 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.53.e_cs
59$D_{4}$ \( 1 + 16 T + 142 T^{2} + 16 p T^{3} + p^{2} T^{4} \) 2.59.q_fm
61$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \) 2.61.au_io
67$D_{4}$ \( 1 + 4 T + 98 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.67.e_du
71$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.71.q_hy
73$D_{4}$ \( 1 - 8 T + 72 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.73.ai_cu
79$D_{4}$ \( 1 - 4 T + 122 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.79.ae_es
83$D_{4}$ \( 1 + 4 T + 160 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.83.e_ge
89$D_{4}$ \( 1 - 20 T + 238 T^{2} - 20 p T^{3} + p^{2} T^{4} \) 2.89.au_je
97$D_{4}$ \( 1 - 4 T + 38 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.97.ae_bm
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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

−9.848443317916173747335658826716, −9.616630476242853064208855805351, −8.937713895707074353855091783864, −8.417730207233797965216599747173, −7.898183885753629869464442637151, −7.895399363381712292026469778154, −7.21738729521407466383350349533, −6.72897918412488033891411690667, −6.25775157059139485414936165440, −6.18375882364950955616071480329, −5.41651533802523318220085879605, −5.35817656475985994720849351478, −4.68428278788750116352676484791, −4.30571643462170437746204485552, −3.57050061363473802858282797451, −3.36080542663436974401221328048, −2.84079256267322368630215735514, −2.52686133606690634051205734907, −1.36980157415612259587668488149, −0.961730809401341751806754824200, 0.961730809401341751806754824200, 1.36980157415612259587668488149, 2.52686133606690634051205734907, 2.84079256267322368630215735514, 3.36080542663436974401221328048, 3.57050061363473802858282797451, 4.30571643462170437746204485552, 4.68428278788750116352676484791, 5.35817656475985994720849351478, 5.41651533802523318220085879605, 6.18375882364950955616071480329, 6.25775157059139485414936165440, 6.72897918412488033891411690667, 7.21738729521407466383350349533, 7.895399363381712292026469778154, 7.898183885753629869464442637151, 8.417730207233797965216599747173, 8.937713895707074353855091783864, 9.616630476242853064208855805351, 9.848443317916173747335658826716

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