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Properties

Label 4-910e2-1.1-c1e2-0-47
Degree $4$
Conductor $828100$
Sign $-1$
Analytic cond. $52.8003$
Root an. cond. $2.69562$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4-s − 2·7-s − 9-s − 4·11-s + 16-s + 12·23-s − 25-s − 2·28-s − 2·29-s − 36-s − 14·37-s + 10·43-s − 4·44-s − 3·49-s − 4·53-s + 2·63-s + 64-s + 8·71-s + 8·77-s + 12·79-s − 8·81-s + 12·92-s + 4·99-s − 100-s − 14·107-s − 2·112-s − 22·113-s + ⋯
L(s)  = 1  + 1/2·4-s − 0.755·7-s − 1/3·9-s − 1.20·11-s + 1/4·16-s + 2.50·23-s − 1/5·25-s − 0.377·28-s − 0.371·29-s − 1/6·36-s − 2.30·37-s + 1.52·43-s − 0.603·44-s − 3/7·49-s − 0.549·53-s + 0.251·63-s + 1/8·64-s + 0.949·71-s + 0.911·77-s + 1.35·79-s − 8/9·81-s + 1.25·92-s + 0.402·99-s − 0.0999·100-s − 1.35·107-s − 0.188·112-s − 2.06·113-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 828100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 828100 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(828100\)    =    \(2^{2} \cdot 5^{2} \cdot 7^{2} \cdot 13^{2}\)
Sign: $-1$
Analytic conductor: \(52.8003\)
Root analytic conductor: \(2.69562\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 828100,\ (\ :1/2, 1/2),\ -1)\)

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_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
5$C_2$ \( 1 + T^{2} \)
7$C_2$ \( 1 + 2 T + p T^{2} \)
13$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
good3$C_2^2$ \( 1 + T^{2} + p^{2} T^{4} \) 2.3.a_b
11$C_2$$\times$$C_2$ \( ( 1 + T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.11.e_z
17$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.17.a_s
19$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \) 2.19.a_abe
23$C_2$$\times$$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 - 5 T + p T^{2} ) \) 2.23.am_dd
29$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.29.c_bi
31$C_2^2$ \( 1 + 45 T^{2} + p^{2} T^{4} \) 2.31.a_bt
37$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \) 2.37.o_et
41$C_2^2$ \( 1 - 33 T^{2} + p^{2} T^{4} \) 2.41.a_abh
43$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) 2.43.ak_dy
47$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) 2.47.a_n
53$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.53.e_ec
59$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \) 2.59.a_abe
61$C_2^2$ \( 1 - 97 T^{2} + p^{2} T^{4} \) 2.61.a_adt
67$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.67.a_ev
71$C_2$$\times$$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.71.ai_cg
73$C_2^2$ \( 1 + 67 T^{2} + p^{2} T^{4} \) 2.73.a_cp
79$C_2$$\times$$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 - T + p T^{2} ) \) 2.79.am_gn
83$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.83.a_co
89$C_2^2$ \( 1 + 54 T^{2} + p^{2} T^{4} \) 2.89.a_cc
97$C_2^2$ \( 1 - 41 T^{2} + p^{2} T^{4} \) 2.97.a_abp
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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

−7.992881185797305431160019180460, −7.51636812434648152641381108695, −7.02498438997078419837665066659, −6.78981803443426067887330344407, −6.30767870571791414609702060861, −5.66535867566320360447350717637, −5.23127031113566843784833958280, −5.06133765066004851543299785259, −4.25085170144088535597182835250, −3.53487795539138272205529151397, −3.11111845996151503180563314740, −2.70716279666795664985811834104, −2.06803164117844889271109136641, −1.12414514387203751094160780039, 0, 1.12414514387203751094160780039, 2.06803164117844889271109136641, 2.70716279666795664985811834104, 3.11111845996151503180563314740, 3.53487795539138272205529151397, 4.25085170144088535597182835250, 5.06133765066004851543299785259, 5.23127031113566843784833958280, 5.66535867566320360447350717637, 6.30767870571791414609702060861, 6.78981803443426067887330344407, 7.02498438997078419837665066659, 7.51636812434648152641381108695, 7.992881185797305431160019180460

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