↑

Properties

Label 4-2448e2-1.1-c1e2-0-17
Degree $4$
Conductor $5992704$
Sign $1$
Analytic cond. $382.100$
Root an. cond. $4.42124$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 4·13-s − 6·17-s − 4·19-s + 8·25-s + 8·43-s + 24·47-s − 4·49-s + 12·53-s − 12·59-s + 20·67-s + 12·83-s + 12·89-s + 36·101-s + 8·103-s + 14·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 14·169-s + 173-s + 179-s + ⋯
L(s)  = 1  + 1.10·13-s − 1.45·17-s − 0.917·19-s + 8/5·25-s + 1.21·43-s + 3.50·47-s − 4/7·49-s + 1.64·53-s − 1.56·59-s + 2.44·67-s + 1.31·83-s + 1.27·89-s + 3.58·101-s + 0.788·103-s + 1.27·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 1.07·169-s + 0.0760·173-s + 0.0747·179-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5992704 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5992704 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(5992704\)    =    \(2^{8} \cdot 3^{4} \cdot 17^{2}\)
Sign: $1$
Analytic conductor: \(382.100\)
Root analytic conductor: \(4.42124\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 5992704,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.721519613\)
\(L(\frac12)\) \(\approx\) \(2.721519613\)
\(L(\frac{3}{2})\) 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 \( 1 \)
17$C_2$ \( 1 + 6 T + p T^{2} \)
good5$C_2^2$ \( 1 - 8 T^{2} + p^{2} T^{4} \) 2.5.a_ai
7$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \) 2.7.a_e
11$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.11.a_ao
13$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.13.ae_be
19$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.19.e_bq
23$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \) 2.23.a_e
29$C_2^2$ \( 1 - 8 T^{2} + p^{2} T^{4} \) 2.29.a_ai
31$C_2^2$ \( 1 - 44 T^{2} + p^{2} T^{4} \) 2.31.a_abs
37$C_2^2$ \( 1 - 56 T^{2} + p^{2} T^{4} \) 2.37.a_ace
41$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \) 2.41.a_aby
43$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.43.ai_dy
47$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \) 2.47.ay_je
53$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.53.am_fm
59$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.59.m_fy
61$C_2^2$ \( 1 + 40 T^{2} + p^{2} T^{4} \) 2.61.a_bo
67$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \) 2.67.au_ja
71$C_2^2$ \( 1 - 140 T^{2} + p^{2} T^{4} \) 2.71.a_afk
73$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.73.a_afq
79$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \) 2.79.a_e
83$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.83.am_hu
89$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.89.am_ig
97$C_2^2$ \( 1 - 122 T^{2} + p^{2} T^{4} \) 2.97.a_aes
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

−9.116400790706771249493848835793, −8.878097428490303369445110910568, −8.508765309248683863561480396399, −8.055835476865620003128115235766, −7.46177454639380369926199832018, −7.32799486940337659590023635773, −6.76155695193498446862346203390, −6.32695470874408879761223608890, −6.20295362000038759166151697578, −5.73088939745887183890801585838, −5.01513058119373375797365995043, −4.92082541919087823250594624402, −4.09853978428856332639384659374, −4.09311962161096485151275963484, −3.52291534832183168968224316919, −2.87080088697272199263018281648, −2.23055697923406186104188172475, −2.16281667347883557240124183661, −1.06891215987451467032544499893, −0.65331502521990335622662412261, 0.65331502521990335622662412261, 1.06891215987451467032544499893, 2.16281667347883557240124183661, 2.23055697923406186104188172475, 2.87080088697272199263018281648, 3.52291534832183168968224316919, 4.09311962161096485151275963484, 4.09853978428856332639384659374, 4.92082541919087823250594624402, 5.01513058119373375797365995043, 5.73088939745887183890801585838, 6.20295362000038759166151697578, 6.32695470874408879761223608890, 6.76155695193498446862346203390, 7.32799486940337659590023635773, 7.46177454639380369926199832018, 8.055835476865620003128115235766, 8.508765309248683863561480396399, 8.878097428490303369445110910568, 9.116400790706771249493848835793

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