Properties

Label 4-3234e2-1.1-c1e2-0-5
Degree $4$
Conductor $10458756$
Sign $1$
Analytic cond. $666.859$
Root an. cond. $5.08169$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s − 2·3-s + 3·4-s + 4·5-s − 4·6-s + 4·8-s + 3·9-s + 8·10-s − 2·11-s − 6·12-s + 8·13-s − 8·15-s + 5·16-s + 4·17-s + 6·18-s + 4·19-s + 12·20-s − 4·22-s − 4·23-s − 8·24-s + 2·25-s + 16·26-s − 4·27-s + 8·29-s − 16·30-s + 4·31-s + 6·32-s + ⋯
L(s)  = 1  + 1.41·2-s − 1.15·3-s + 3/2·4-s + 1.78·5-s − 1.63·6-s + 1.41·8-s + 9-s + 2.52·10-s − 0.603·11-s − 1.73·12-s + 2.21·13-s − 2.06·15-s + 5/4·16-s + 0.970·17-s + 1.41·18-s + 0.917·19-s + 2.68·20-s − 0.852·22-s − 0.834·23-s − 1.63·24-s + 2/5·25-s + 3.13·26-s − 0.769·27-s + 1.48·29-s − 2.92·30-s + 0.718·31-s + 1.06·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 10458756 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10458756 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(10458756\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{4} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(666.859\)
Root analytic conductor: \(5.08169\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{3234} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 10458756,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(9.980286780\)
\(L(\frac12)\) \(\approx\) \(9.980286780\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 + T )^{2} \)
7 \( 1 \)
11$C_1$ \( ( 1 + T )^{2} \)
good5$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
13$D_{4}$ \( 1 - 8 T + 40 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
17$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
19$D_{4}$ \( 1 - 4 T + 24 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
23$C_4$ \( 1 + 4 T + 42 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 - 8 T + 42 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 - 4 T + 48 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 + 4 T + 70 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
41$C_4$ \( 1 - 12 T + 86 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 + 4 T + 82 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 - 4 T - 4 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 + 4 T + 38 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 - 8 T + 126 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 - 8 T + 136 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 + 8 T + 142 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
71$C_2^2$ \( 1 + 110 T^{2} + p^{2} T^{4} \)
73$D_{4}$ \( 1 - 12 T + 150 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 + 16 T + 150 T^{2} + 16 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 + 4 T + 120 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 - 16 T + 224 T^{2} - 16 p T^{3} + p^{2} T^{4} \)
97$C_2^2$ \( 1 + 96 T^{2} + p^{2} T^{4} \)
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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

−8.730221683107925808781596054396, −8.452700009455326605632775384571, −7.981469552544518552284372504302, −7.60736417899162859929339382891, −7.07901918283665506212307628651, −6.73738043986355651898002442427, −6.17550687159741192834472047394, −6.05734371665512115240384388632, −5.80831539177917704294157620685, −5.64025977602765499140288703340, −5.07721947019006504851002132502, −4.85853113041344073102923490752, −4.11552285298945100391736118888, −3.99345345914965449238724140847, −3.15924126640616148548450260567, −3.11308842732284167142129486182, −2.07382726814357683375991202659, −2.06646542973630044305340683908, −1.05504638080848840359357740156, −1.05303966768808338989353271016, 1.05303966768808338989353271016, 1.05504638080848840359357740156, 2.06646542973630044305340683908, 2.07382726814357683375991202659, 3.11308842732284167142129486182, 3.15924126640616148548450260567, 3.99345345914965449238724140847, 4.11552285298945100391736118888, 4.85853113041344073102923490752, 5.07721947019006504851002132502, 5.64025977602765499140288703340, 5.80831539177917704294157620685, 6.05734371665512115240384388632, 6.17550687159741192834472047394, 6.73738043986355651898002442427, 7.07901918283665506212307628651, 7.60736417899162859929339382891, 7.981469552544518552284372504302, 8.452700009455326605632775384571, 8.730221683107925808781596054396

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