Properties

Label 4-8470e2-1.1-c1e2-0-18
Degree $4$
Conductor $71740900$
Sign $1$
Analytic cond. $4574.26$
Root an. cond. $8.22394$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s + 3·4-s − 2·5-s + 2·7-s + 4·8-s − 3·9-s − 4·10-s − 8·13-s + 4·14-s + 5·16-s + 2·17-s − 6·18-s − 4·19-s − 6·20-s − 10·23-s + 3·25-s − 16·26-s + 6·28-s + 16·29-s − 2·31-s + 6·32-s + 4·34-s − 4·35-s − 9·36-s + 12·37-s − 8·38-s − 8·40-s + ⋯
L(s)  = 1  + 1.41·2-s + 3/2·4-s − 0.894·5-s + 0.755·7-s + 1.41·8-s − 9-s − 1.26·10-s − 2.21·13-s + 1.06·14-s + 5/4·16-s + 0.485·17-s − 1.41·18-s − 0.917·19-s − 1.34·20-s − 2.08·23-s + 3/5·25-s − 3.13·26-s + 1.13·28-s + 2.97·29-s − 0.359·31-s + 1.06·32-s + 0.685·34-s − 0.676·35-s − 3/2·36-s + 1.97·37-s − 1.29·38-s − 1.26·40-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 71740900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 71740900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(71740900\)    =    \(2^{2} \cdot 5^{2} \cdot 7^{2} \cdot 11^{4}\)
Sign: $1$
Analytic conductor: \(4574.26\)
Root analytic conductor: \(8.22394\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{8470} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 71740900,\ (\ :1/2, 1/2),\ 1)\)

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)$
bad2$C_1$ \( ( 1 - T )^{2} \)
5$C_1$ \( ( 1 + T )^{2} \)
7$C_1$ \( ( 1 - T )^{2} \)
11 \( 1 \)
good3$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \)
13$D_{4}$ \( 1 + 8 T + 3 p T^{2} + 8 p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 - 2 T + 32 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 + 4 T + 39 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
23$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
31$D_{4}$ \( 1 + 2 T + 60 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
37$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
41$D_{4}$ \( 1 + 2 T + 56 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 + 14 T + 132 T^{2} + 14 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 + 6 T + 100 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 + 2 T + 80 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 + 18 T + 187 T^{2} + 18 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 - 8 T + 90 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 + 2 T + 60 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
71$C_2^2$ \( 1 + 94 T^{2} + p^{2} T^{4} \)
73$D_{4}$ \( 1 + 6 T + 8 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
79$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
89$D_{4}$ \( 1 + 10 T + 128 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 - 8 T + 162 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
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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

−7.57656453329559653650924683028, −7.25597808506703082965099090183, −6.81566654989916828222344949127, −6.56473863487662978007748178725, −6.07469671894713726896476165327, −5.98295481352473436318050127857, −5.35856665347773804368576105919, −5.08973897601132819678795815427, −4.62261579920303810979909211581, −4.62192240377085797763275861384, −4.20288872172410691753366818714, −3.84762583098979055230826690989, −3.12722255141364512205979452172, −3.02590800743813573302839594135, −2.48505100324600491843778179018, −2.35974226910059170901582185345, −1.65999264438147676987680905520, −1.17388544740102819984974445059, 0, 0, 1.17388544740102819984974445059, 1.65999264438147676987680905520, 2.35974226910059170901582185345, 2.48505100324600491843778179018, 3.02590800743813573302839594135, 3.12722255141364512205979452172, 3.84762583098979055230826690989, 4.20288872172410691753366818714, 4.62192240377085797763275861384, 4.62261579920303810979909211581, 5.08973897601132819678795815427, 5.35856665347773804368576105919, 5.98295481352473436318050127857, 6.07469671894713726896476165327, 6.56473863487662978007748178725, 6.81566654989916828222344949127, 7.25597808506703082965099090183, 7.57656453329559653650924683028

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