↑

Properties

Label 4-4080e2-1.1-c1e2-0-26
Degree $4$
Conductor $16646400$
Sign $1$
Analytic cond. $1061.38$
Root an. cond. $5.70779$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 9-s − 4·13-s − 2·17-s − 16·19-s − 25-s + 10·49-s − 12·53-s + 24·59-s + 8·67-s + 81-s − 32·83-s − 28·89-s + 20·101-s − 16·103-s + 4·117-s + 22·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 2·153-s + 157-s + 163-s + 167-s − 14·169-s + ⋯
L(s)  = 1  − 1/3·9-s − 1.10·13-s − 0.485·17-s − 3.67·19-s − 1/5·25-s + 10/7·49-s − 1.64·53-s + 3.12·59-s + 0.977·67-s + 1/9·81-s − 3.51·83-s − 2.96·89-s + 1.99·101-s − 1.57·103-s + 0.369·117-s + 2·121-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 + 0.161·153-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 1.07·169-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 16646400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16646400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(16646400\)    =    \(2^{8} \cdot 3^{2} \cdot 5^{2} \cdot 17^{2}\)
Sign: $1$
Analytic conductor: \(1061.38\)
Root analytic conductor: \(5.70779\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 16646400,\ (\ :1/2, 1/2),\ 1)\)

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 \( 1 \)
3$C_2$ \( 1 + T^{2} \)
5$C_2$ \( 1 + T^{2} \)
17$C_2$ \( 1 + 2 T + p T^{2} \)
good7$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.7.a_ak
11$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.11.a_aw
13$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.13.e_be
19$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.19.q_dy
23$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.23.a_ak
29$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.29.a_bq
31$C_2^2$ \( 1 - 58 T^{2} + p^{2} T^{4} \) 2.31.a_acg
37$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.37.a_acs
41$C_2^2$ \( 1 + 62 T^{2} + p^{2} T^{4} \) 2.41.a_ck
43$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.43.a_di
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$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \) 2.59.ay_kc
61$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.61.a_aw
67$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.67.ai_fu
71$C_2^2$ \( 1 - 138 T^{2} + p^{2} T^{4} \) 2.71.a_afi
73$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \) 2.73.a_afa
79$C_2^2$ \( 1 - 58 T^{2} + p^{2} T^{4} \) 2.79.a_acg
83$C_2$ \( ( 1 + 16 T + p T^{2} )^{2} \) 2.83.bg_qg
89$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \) 2.89.bc_ok
97$C_2$ \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) 2.97.a_afa
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

−8.394562033659729179225543733977, −7.951832757406100505317239313977, −7.55984931101520415403679010933, −6.93114410016101673652223149239, −6.79766786767598238341282463884, −6.62198036048398191905015900192, −5.94702995924423706252537110956, −5.72253328100391250226742543971, −5.34406963745896250282040846276, −4.74503872086412944767497593268, −4.30828969454291751852009096924, −4.24619542465025733406706073926, −3.77370844892674580985622698297, −3.07998933056709285314319634509, −2.54613287370627998424943012928, −2.20576110230163555304284829850, −1.99605261998539542792446719571, −1.10686928652259138820908802265, 0, 0, 1.10686928652259138820908802265, 1.99605261998539542792446719571, 2.20576110230163555304284829850, 2.54613287370627998424943012928, 3.07998933056709285314319634509, 3.77370844892674580985622698297, 4.24619542465025733406706073926, 4.30828969454291751852009096924, 4.74503872086412944767497593268, 5.34406963745896250282040846276, 5.72253328100391250226742543971, 5.94702995924423706252537110956, 6.62198036048398191905015900192, 6.79766786767598238341282463884, 6.93114410016101673652223149239, 7.55984931101520415403679010933, 7.951832757406100505317239313977, 8.394562033659729179225543733977

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