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Properties

Label 4-980e2-1.1-c1e2-0-12
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 $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·5-s + 6·9-s + 8·19-s − 25-s − 4·29-s + 16·31-s − 12·41-s − 12·45-s − 8·59-s + 12·61-s + 24·71-s + 8·79-s + 27·81-s − 20·89-s − 16·95-s + 36·101-s − 28·109-s − 22·121-s + 12·125-s + 127-s + 131-s + 137-s + 139-s + 8·145-s + 149-s + 151-s − 32·155-s + ⋯
L(s)  = 1  − 0.894·5-s + 2·9-s + 1.83·19-s − 1/5·25-s − 0.742·29-s + 2.87·31-s − 1.87·41-s − 1.78·45-s − 1.04·59-s + 1.53·61-s + 2.84·71-s + 0.900·79-s + 3·81-s − 2.11·89-s − 1.64·95-s + 3.58·101-s − 2.68·109-s − 2·121-s + 1.07·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.664·145-s + 0.0819·149-s + 0.0813·151-s − 2.57·155-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: \(0\)
Selberg data: \((4,\ 960400,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.275759636\)
\(L(\frac12)\) \(\approx\) \(2.275759636\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + 2 T + p T^{2} \)
7 \( 1 \)
good3$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.3.a_ag
11$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.11.a_w
13$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.13.a_ak
17$C_2^2$ \( 1 - 18 T^{2} + p^{2} T^{4} \) 2.17.a_as
19$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.19.ai_cc
23$C_2^2$ \( 1 + 18 T^{2} + p^{2} T^{4} \) 2.23.a_s
29$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.29.e_ck
31$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.31.aq_ew
37$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.37.a_ak
41$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.41.m_eo
43$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \) 2.43.a_aw
47$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \) 2.47.a_abe
53$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.53.a_aec
59$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.59.i_fe
61$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.61.am_gc
67$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \) 2.67.a_acs
71$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \) 2.71.ay_la
73$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \) 2.73.a_afa
79$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.79.ai_gs
83$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.83.a_agk
89$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.89.u_ks
97$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \) 2.97.a_aby
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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

−10.05233075025307437457255620931, −9.718041539332464626372441339389, −9.664454064208399134561197183584, −9.064975301251896972903238645405, −8.257272721551026313630727910791, −8.142734267157461736751636478283, −7.73245463829029371021455980558, −7.26938529790489518941486150390, −6.82286378000839713123443864296, −6.66267580602216179562967404843, −5.97249057170617375311967569198, −5.27094420487273348104079897818, −4.85222644296536460126174026308, −4.55558651605682777273395501259, −3.75797069881104069637612385180, −3.71245667115707378202967247638, −2.97490547179958909482585564847, −2.17543261314394075770418611888, −1.38123482634235965320934201771, −0.795800109119008986207509747722, 0.795800109119008986207509747722, 1.38123482634235965320934201771, 2.17543261314394075770418611888, 2.97490547179958909482585564847, 3.71245667115707378202967247638, 3.75797069881104069637612385180, 4.55558651605682777273395501259, 4.85222644296536460126174026308, 5.27094420487273348104079897818, 5.97249057170617375311967569198, 6.66267580602216179562967404843, 6.82286378000839713123443864296, 7.26938529790489518941486150390, 7.73245463829029371021455980558, 8.142734267157461736751636478283, 8.257272721551026313630727910791, 9.064975301251896972903238645405, 9.664454064208399134561197183584, 9.718041539332464626372441339389, 10.05233075025307437457255620931

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