Properties

Label 4-4830e2-1.1-c1e2-0-6
Degree $4$
Conductor $23328900$
Sign $1$
Analytic cond. $1487.47$
Root an. cond. $6.21029$
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 + 2·3-s + 3·4-s − 2·5-s − 4·6-s − 2·7-s − 4·8-s + 3·9-s + 4·10-s − 4·11-s + 6·12-s + 2·13-s + 4·14-s − 4·15-s + 5·16-s + 2·17-s − 6·18-s + 2·19-s − 6·20-s − 4·21-s + 8·22-s + 2·23-s − 8·24-s + 3·25-s − 4·26-s + 4·27-s − 6·28-s + ⋯
L(s)  = 1  − 1.41·2-s + 1.15·3-s + 3/2·4-s − 0.894·5-s − 1.63·6-s − 0.755·7-s − 1.41·8-s + 9-s + 1.26·10-s − 1.20·11-s + 1.73·12-s + 0.554·13-s + 1.06·14-s − 1.03·15-s + 5/4·16-s + 0.485·17-s − 1.41·18-s + 0.458·19-s − 1.34·20-s − 0.872·21-s + 1.70·22-s + 0.417·23-s − 1.63·24-s + 3/5·25-s − 0.784·26-s + 0.769·27-s − 1.13·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 23328900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23328900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(23328900\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 7^{2} \cdot 23^{2}\)
Sign: $1$
Analytic conductor: \(1487.47\)
Root analytic conductor: \(6.21029\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{4830} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 23328900,\ (\ :1/2, 1/2),\ 1)\)

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$C_1$ \( ( 1 + T )^{2} \)
3$C_1$ \( ( 1 - T )^{2} \)
5$C_1$ \( ( 1 + T )^{2} \)
7$C_1$ \( ( 1 + T )^{2} \)
23$C_1$ \( ( 1 - T )^{2} \)
good11$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
13$D_{4}$ \( 1 - 2 T + 14 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 - 2 T + 22 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 - 2 T + 26 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 + 4 T + 10 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 + 2 T + 50 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
37$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
43$C_4$ \( 1 - 8 T + 50 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 + 10 T + 106 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 - 8 T + 70 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 - 4 T + 70 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
61$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
67$D_{4}$ \( 1 + 8 T + 98 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
79$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \)
83$D_{4}$ \( 1 + 14 T + 202 T^{2} + 14 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 + 18 T + 246 T^{2} + 18 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 + 2 T + 78 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
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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

−8.039919680204466614987862539193, −7.997221276645637333862073764183, −7.32351862606737712027502723467, −7.30201679938012644429355609637, −6.80078597075796584936728177354, −6.79001400608949395894850163636, −5.81654082660843496525887630050, −5.79777046659397449562374423228, −5.05692550796154376312745412987, −4.95296779091825928960110423220, −4.01770057736007391864931177361, −3.71778637726064009816623969745, −3.43559257132793337111499627761, −3.05916569537052200155536694523, −2.51678768453414848261967850860, −2.31284647615693733289848997715, −1.35040915657556887100825302783, −1.28565551540535630306896476209, 0, 0, 1.28565551540535630306896476209, 1.35040915657556887100825302783, 2.31284647615693733289848997715, 2.51678768453414848261967850860, 3.05916569537052200155536694523, 3.43559257132793337111499627761, 3.71778637726064009816623969745, 4.01770057736007391864931177361, 4.95296779091825928960110423220, 5.05692550796154376312745412987, 5.79777046659397449562374423228, 5.81654082660843496525887630050, 6.79001400608949395894850163636, 6.80078597075796584936728177354, 7.30201679938012644429355609637, 7.32351862606737712027502723467, 7.997221276645637333862073764183, 8.039919680204466614987862539193

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