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Properties

Label 4-960e2-1.1-c1e2-0-32
Degree $4$
Conductor $921600$
Sign $1$
Analytic cond. $58.7620$
Root an. cond. $2.76868$
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 + 9-s − 4·13-s + 12·17-s + 3·25-s + 12·29-s − 4·37-s − 12·41-s + 2·45-s + 2·49-s + 12·53-s + 20·61-s − 8·65-s + 4·73-s + 81-s + 24·85-s + 36·89-s + 4·97-s − 36·101-s + 20·109-s − 36·113-s − 4·117-s − 22·121-s + 4·125-s + 127-s + 131-s + 137-s + ⋯
L(s)  = 1  + 0.894·5-s + 1/3·9-s − 1.10·13-s + 2.91·17-s + 3/5·25-s + 2.22·29-s − 0.657·37-s − 1.87·41-s + 0.298·45-s + 2/7·49-s + 1.64·53-s + 2.56·61-s − 0.992·65-s + 0.468·73-s + 1/9·81-s + 2.60·85-s + 3.81·89-s + 0.406·97-s − 3.58·101-s + 1.91·109-s − 3.38·113-s − 0.369·117-s − 2·121-s + 0.357·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 921600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 921600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(921600\)    =    \(2^{12} \cdot 3^{2} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(58.7620\)
Root analytic conductor: \(2.76868\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 921600,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.912152815\)
\(L(\frac12)\) \(\approx\) \(2.912152815\)
\(L(\frac{3}{2})\) 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_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
5$C_1$ \( ( 1 - T )^{2} \)
good7$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.7.a_ac
11$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.11.a_w
13$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.13.e_be
17$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.17.am_cs
19$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.19.a_w
23$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.23.a_bu
29$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.29.am_dq
31$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.31.a_ac
37$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.37.e_da
41$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.41.m_eo
43$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.43.a_cs
47$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.47.a_dq
53$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.53.am_fm
59$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.59.a_eo
61$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \) 2.61.au_io
67$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.67.a_eo
71$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.71.a_fm
73$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.73.ae_fu
79$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.79.a_dq
83$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.83.a_w
89$C_2$ \( ( 1 - 18 T + p T^{2} )^{2} \) 2.89.abk_ti
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

−8.307525687782695431948900879973, −7.71052407307967077644420749342, −7.26843774815325716692994821945, −6.76895238282715006434981339956, −6.55369649177171482035005226183, −5.77106186730486543398971536544, −5.48025531888955534300415720953, −4.95941068749140901060295823906, −4.84143328212506167706839183620, −3.69936628544405197022333195268, −3.58551752676900090223959184299, −2.69384090145739289224125918540, −2.39041205999551187286748903960, −1.43294542786364492056174526795, −0.889333176383366415114391202530, 0.889333176383366415114391202530, 1.43294542786364492056174526795, 2.39041205999551187286748903960, 2.69384090145739289224125918540, 3.58551752676900090223959184299, 3.69936628544405197022333195268, 4.84143328212506167706839183620, 4.95941068749140901060295823906, 5.48025531888955534300415720953, 5.77106186730486543398971536544, 6.55369649177171482035005226183, 6.76895238282715006434981339956, 7.26843774815325716692994821945, 7.71052407307967077644420749342, 8.307525687782695431948900879973

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