Properties

Label 4-3174e2-1.1-c1e2-0-8
Degree $4$
Conductor $10074276$
Sign $1$
Analytic cond. $642.344$
Root an. cond. $5.03433$
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 + 2·5-s + 4·6-s + 4·8-s + 3·9-s + 4·10-s + 6·11-s + 6·12-s + 4·15-s + 5·16-s + 8·17-s + 6·18-s + 2·19-s + 6·20-s + 12·22-s + 8·24-s − 2·25-s + 4·27-s + 8·30-s + 4·31-s + 6·32-s + 12·33-s + 16·34-s + 9·36-s − 18·37-s + ⋯
L(s)  = 1  + 1.41·2-s + 1.15·3-s + 3/2·4-s + 0.894·5-s + 1.63·6-s + 1.41·8-s + 9-s + 1.26·10-s + 1.80·11-s + 1.73·12-s + 1.03·15-s + 5/4·16-s + 1.94·17-s + 1.41·18-s + 0.458·19-s + 1.34·20-s + 2.55·22-s + 1.63·24-s − 2/5·25-s + 0.769·27-s + 1.46·30-s + 0.718·31-s + 1.06·32-s + 2.08·33-s + 2.74·34-s + 3/2·36-s − 2.95·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 10074276 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10074276 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(10074276\)    =    \(2^{2} \cdot 3^{2} \cdot 23^{4}\)
Sign: $1$
Analytic conductor: \(642.344\)
Root analytic conductor: \(5.03433\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 10074276,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(19.65790953\)
\(L(\frac12)\) \(\approx\) \(19.65790953\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 - T )^{2} \)
23 \( 1 \)
good5$D_{4}$ \( 1 - 2 T + 6 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
7$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \)
11$C_4$ \( 1 - 6 T + 26 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
13$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \)
17$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
19$C_4$ \( 1 - 2 T - 6 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
29$C_2^2$ \( 1 + 38 T^{2} + p^{2} T^{4} \)
31$C_4$ \( 1 - 4 T + 46 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 + 18 T + 150 T^{2} + 18 p T^{3} + p^{2} T^{4} \)
41$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
43$D_{4}$ \( 1 - 14 T + 130 T^{2} - 14 p T^{3} + p^{2} T^{4} \)
47$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
53$D_{4}$ \( 1 + 6 T + 110 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 + 38 T^{2} + p^{2} T^{4} \)
61$D_{4}$ \( 1 + 6 T + 126 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 + 6 T + 98 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
71$C_2^2$ \( 1 + 62 T^{2} + p^{2} T^{4} \)
73$C_2^2$ \( 1 + 126 T^{2} + p^{2} T^{4} \)
79$C_2^2$ \( 1 + 138 T^{2} + p^{2} T^{4} \)
83$D_{4}$ \( 1 + 22 T + 282 T^{2} + 22 p T^{3} + p^{2} T^{4} \)
89$C_4$ \( 1 - 12 T + 194 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 - 8 T + 190 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
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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

−8.961962908553550004489310627762, −8.646515706565953878841644923077, −7.88077482279980279661801468170, −7.68009652823819855391609531230, −7.31049354114779203849974613491, −7.00740905512036870265019794104, −6.35508843535325978501003240906, −6.33977126711634566005120384806, −5.62210841135063879684587788492, −5.58681294003665807559192020104, −5.00481801069111586271087232843, −4.54918994635621799668434980638, −3.96099586854095752007575502368, −3.75483423974099476760505675854, −3.43654483710125286533570535496, −2.91139738691773074664742373955, −2.51716272798110563496820644548, −1.89303614735994328170303462046, −1.39414480973934441384358648368, −1.15059298584104255275817229374, 1.15059298584104255275817229374, 1.39414480973934441384358648368, 1.89303614735994328170303462046, 2.51716272798110563496820644548, 2.91139738691773074664742373955, 3.43654483710125286533570535496, 3.75483423974099476760505675854, 3.96099586854095752007575502368, 4.54918994635621799668434980638, 5.00481801069111586271087232843, 5.58681294003665807559192020104, 5.62210841135063879684587788492, 6.33977126711634566005120384806, 6.35508843535325978501003240906, 7.00740905512036870265019794104, 7.31049354114779203849974613491, 7.68009652823819855391609531230, 7.88077482279980279661801468170, 8.646515706565953878841644923077, 8.961962908553550004489310627762

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