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Properties

Label 4-980e2-1.1-c1e2-0-31
Degree $4$
Conductor $960400$
Sign $1$
Analytic cond. $61.2359$
Root an. cond. $2.79738$
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  − 2·3-s + 2·5-s − 9-s − 2·11-s − 10·13-s − 4·15-s − 10·17-s + 4·19-s + 4·23-s + 3·25-s + 6·27-s + 2·29-s − 12·31-s + 4·33-s − 4·37-s + 20·39-s − 4·41-s + 12·43-s − 2·45-s − 2·47-s + 20·51-s − 16·53-s − 4·55-s − 8·57-s − 4·59-s − 16·61-s − 20·65-s + ⋯
L(s)  = 1  − 1.15·3-s + 0.894·5-s − 1/3·9-s − 0.603·11-s − 2.77·13-s − 1.03·15-s − 2.42·17-s + 0.917·19-s + 0.834·23-s + 3/5·25-s + 1.15·27-s + 0.371·29-s − 2.15·31-s + 0.696·33-s − 0.657·37-s + 3.20·39-s − 0.624·41-s + 1.82·43-s − 0.298·45-s − 0.291·47-s + 2.80·51-s − 2.19·53-s − 0.539·55-s − 1.05·57-s − 0.520·59-s − 2.04·61-s − 2.48·65-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 960400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(960400\)    =    \(2^{4} \cdot 5^{2} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(61.2359\)
Root analytic conductor: \(2.79738\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 960400,\ (\ :1/2, 1/2),\ 1)\)

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 \( 1 \)
5$C_1$ \( ( 1 - T )^{2} \)
7 \( 1 \)
good3$D_{4}$ \( 1 + 2 T + 5 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.3.c_f
11$D_{4}$ \( 1 + 2 T + 15 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.11.c_p
13$D_{4}$ \( 1 + 10 T + 49 T^{2} + 10 p T^{3} + p^{2} T^{4} \) 2.13.k_bx
17$D_{4}$ \( 1 + 10 T + 57 T^{2} + 10 p T^{3} + p^{2} T^{4} \) 2.17.k_cf
19$D_{4}$ \( 1 - 4 T + 10 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.19.ae_k
23$D_{4}$ \( 1 - 4 T + 48 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.23.ae_bw
29$D_{4}$ \( 1 - 2 T + 27 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.29.ac_bb
31$D_{4}$ \( 1 + 12 T + 96 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.31.m_ds
37$D_{4}$ \( 1 + 4 T + 76 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.37.e_cy
41$D_{4}$ \( 1 + 4 T + 84 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.41.e_dg
43$D_{4}$ \( 1 - 12 T + 90 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.43.am_dm
47$D_{4}$ \( 1 + 2 T - 3 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.47.c_ad
53$D_{4}$ \( 1 + 16 T + 152 T^{2} + 16 p T^{3} + p^{2} T^{4} \) 2.53.q_fw
59$D_{4}$ \( 1 + 4 T + 120 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.59.e_eq
61$D_{4}$ \( 1 + 16 T + 178 T^{2} + 16 p T^{3} + p^{2} T^{4} \) 2.61.q_gw
67$D_{4}$ \( 1 + 8 T + 100 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.67.i_dw
71$D_{4}$ \( 1 + 4 T + 74 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.71.e_cw
73$D_{4}$ \( 1 + 16 T + 202 T^{2} + 16 p T^{3} + p^{2} T^{4} \) 2.73.q_hu
79$D_{4}$ \( 1 + 2 T - 41 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.79.c_abp
83$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.83.aq_iw
89$C_2^2$ \( 1 - 110 T^{2} + p^{2} T^{4} \) 2.89.a_aeg
97$D_{4}$ \( 1 + 6 T + 41 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.97.g_bp
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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

−9.631238679195046345978330328254, −9.436586401103624970366121579536, −9.102691490295267057013913864365, −8.817887005457252545901925322918, −8.001682359547254971705465851892, −7.54095536878199713415402242985, −7.07028403257897782882513009419, −6.97627950513349276641135508582, −6.10348288756034816076782637821, −6.05815683316642060527335324338, −5.22370344588509756500589218610, −5.18558606489697503301758832989, −4.69839196630239099295080704262, −4.34560793882724563776534950263, −3.12718643944308874880015815755, −2.79899269739276673639862887077, −2.24009351742173665139724222091, −1.63176368754881663258788704565, 0, 0, 1.63176368754881663258788704565, 2.24009351742173665139724222091, 2.79899269739276673639862887077, 3.12718643944308874880015815755, 4.34560793882724563776534950263, 4.69839196630239099295080704262, 5.18558606489697503301758832989, 5.22370344588509756500589218610, 6.05815683316642060527335324338, 6.10348288756034816076782637821, 6.97627950513349276641135508582, 7.07028403257897782882513009419, 7.54095536878199713415402242985, 8.001682359547254971705465851892, 8.817887005457252545901925322918, 9.102691490295267057013913864365, 9.436586401103624970366121579536, 9.631238679195046345978330328254

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