Properties

Label 4-308e2-1.1-c1e2-0-13
Degree $4$
Conductor $94864$
Sign $-1$
Analytic cond. $6.04861$
Root an. cond. $1.56824$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $1$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2·4-s + 6·5-s − 5·9-s − 8·13-s + 4·16-s − 12·17-s − 12·20-s + 17·25-s − 12·29-s + 10·36-s + 22·37-s + 12·41-s − 30·45-s + 49-s + 16·52-s − 12·53-s − 20·61-s − 8·64-s − 48·65-s + 24·68-s + 4·73-s + 24·80-s + 16·81-s − 72·85-s − 6·89-s − 2·97-s − 34·100-s + ⋯
L(s)  = 1  − 4-s + 2.68·5-s − 5/3·9-s − 2.21·13-s + 16-s − 2.91·17-s − 2.68·20-s + 17/5·25-s − 2.22·29-s + 5/3·36-s + 3.61·37-s + 1.87·41-s − 4.47·45-s + 1/7·49-s + 2.21·52-s − 1.64·53-s − 2.56·61-s − 64-s − 5.95·65-s + 2.91·68-s + 0.468·73-s + 2.68·80-s + 16/9·81-s − 7.80·85-s − 0.635·89-s − 0.203·97-s − 3.39·100-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 94864 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 94864 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(94864\)    =    \(2^{4} \cdot 7^{2} \cdot 11^{2}\)
Sign: $-1$
Analytic conductor: \(6.04861\)
Root analytic conductor: \(1.56824\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 94864,\ (\ :1/2, 1/2),\ -1)\)

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$C_2$ \( 1 + p T^{2} \)
7$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
11$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
good3$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) 2.3.a_f
5$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \) 2.5.ag_t
13$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.13.i_bq
17$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.17.m_cs
19$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.19.a_bi
23$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.23.a_bl
29$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.29.m_dq
31$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.31.a_bl
37$C_2$ \( ( 1 - 11 T + p T^{2} )^{2} \) 2.37.aw_hn
41$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.41.am_eo
43$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.43.a_w
47$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.47.a_dq
53$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.53.m_fm
59$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) 2.59.a_bl
61$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.61.u_io
67$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.67.a_ef
71$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) 2.71.a_cj
73$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.73.ae_fu
79$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.79.a_cg
83$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.83.a_w
89$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \) 2.89.g_hf
97$C_2$ \( ( 1 + T + p T^{2} )^{2} \) 2.97.c_hn
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

−9.410643014656771968003304964204, −9.121939609810073781754575230776, −8.765104139486944533319691232801, −7.79543673470654027074280114997, −7.53612866230739269099963349548, −6.50137819226683740639555041041, −6.03143936736094879672909382578, −5.89237731397243587720148634695, −5.23765283586375285456916174906, −4.69940224529243356978622440045, −4.26207385712577519938962673430, −2.73562708382873437540236845199, −2.52489627324288051568453078479, −1.92362259290157844533526668953, 0, 1.92362259290157844533526668953, 2.52489627324288051568453078479, 2.73562708382873437540236845199, 4.26207385712577519938962673430, 4.69940224529243356978622440045, 5.23765283586375285456916174906, 5.89237731397243587720148634695, 6.03143936736094879672909382578, 6.50137819226683740639555041041, 7.53612866230739269099963349548, 7.79543673470654027074280114997, 8.765104139486944533319691232801, 9.121939609810073781754575230776, 9.410643014656771968003304964204

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