↑

Properties

Label 4-387200-1.1-c1e2-0-13
Degree $4$
Conductor $387200$
Sign $1$
Analytic cond. $24.6882$
Root an. cond. $2.22906$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2·5-s + 4·7-s + 2·9-s + 8·19-s + 3·25-s + 8·35-s − 4·37-s + 4·43-s + 4·45-s + 2·49-s + 12·53-s + 8·63-s − 5·81-s − 12·83-s + 4·89-s + 16·95-s − 12·97-s − 4·107-s − 28·113-s − 11·121-s + 4·125-s + 127-s + 131-s + 32·133-s + 137-s + 139-s + 149-s + ⋯
L(s)  = 1  + 0.894·5-s + 1.51·7-s + 2/3·9-s + 1.83·19-s + 3/5·25-s + 1.35·35-s − 0.657·37-s + 0.609·43-s + 0.596·45-s + 2/7·49-s + 1.64·53-s + 1.00·63-s − 5/9·81-s − 1.31·83-s + 0.423·89-s + 1.64·95-s − 1.21·97-s − 0.386·107-s − 2.63·113-s − 121-s + 0.357·125-s + 0.0887·127-s + 0.0873·131-s + 2.77·133-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 387200 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 387200 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(387200\)    =    \(2^{7} \cdot 5^{2} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(24.6882\)
Root analytic conductor: \(2.22906\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 387200,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.159897137\)
\(L(\frac12)\) \(\approx\) \(3.159897137\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 - T )^{2} \)
11$C_2$ \( 1 + p T^{2} \)
good3$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.3.a_ac
7$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) 2.7.ae_o
13$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \) 2.13.a_o
17$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \) 2.17.a_g
19$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) 2.19.ai_bm
23$C_2^2$ \( 1 + 22 T^{2} + p^{2} T^{4} \) 2.23.a_w
29$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.29.a_ak
31$C_2^2$ \( 1 - 18 T^{2} + p^{2} T^{4} \) 2.31.a_as
37$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.37.e_o
41$C_2^2$ \( 1 + 30 T^{2} + p^{2} T^{4} \) 2.41.a_be
43$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.43.ae_cc
47$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \) 2.47.a_g
53$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.53.am_fm
59$C_2^2$ \( 1 - 90 T^{2} + p^{2} T^{4} \) 2.59.a_adm
61$C_2^2$ \( 1 - 26 T^{2} + p^{2} T^{4} \) 2.61.a_aba
67$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \) 2.67.a_abi
71$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \) 2.71.a_aby
73$C_2^2$ \( 1 - 106 T^{2} + p^{2} T^{4} \) 2.73.a_aec
79$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.79.a_fm
83$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.83.m_gk
89$C_2$$\times$$C_2$ \( ( 1 - 18 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.89.ae_acw
97$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.97.m_gk
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.618682628665420280996220498022, −8.240222389730593457964628653362, −7.70200207293889953687826754199, −7.32391724627428462753174682459, −6.93982858233685330930908381046, −6.33583797650597465447621281920, −5.67024949903968962367204137625, −5.24994886271652254981768200541, −5.06084578965330143300269549972, −4.31158463954656220302583301841, −3.85309381603312166996610685441, −2.98808210810482496056029083883, −2.37921741827803489995585398947, −1.54617278880287985105468381852, −1.18789081324829946125149839911, 1.18789081324829946125149839911, 1.54617278880287985105468381852, 2.37921741827803489995585398947, 2.98808210810482496056029083883, 3.85309381603312166996610685441, 4.31158463954656220302583301841, 5.06084578965330143300269549972, 5.24994886271652254981768200541, 5.67024949903968962367204137625, 6.33583797650597465447621281920, 6.93982858233685330930908381046, 7.32391724627428462753174682459, 7.70200207293889953687826754199, 8.240222389730593457964628653362, 8.618682628665420280996220498022

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