Properties

Label 4-75e2-1.1-c8e2-0-0
Degree $4$
Conductor $5625$
Sign $1$
Analytic cond. $933.509$
Root an. cond. $5.52751$
Motivic weight $8$
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  − 90·3-s + 8·4-s + 3.50e3·7-s + 1.53e3·9-s − 720·12-s − 5.14e4·13-s − 6.54e4·16-s + 3.78e4·19-s − 3.15e5·21-s + 4.51e5·27-s + 2.80e4·28-s − 7.02e5·31-s + 1.23e4·36-s − 2.67e6·37-s + 4.63e6·39-s + 7.05e6·43-s + 5.89e6·48-s − 2.34e6·49-s − 4.11e5·52-s − 3.40e6·57-s + 1.50e6·61-s + 5.38e6·63-s − 1.04e6·64-s − 4.53e6·67-s − 5.53e7·73-s + 3.03e5·76-s − 4.59e7·79-s + ⋯
L(s)  = 1  − 1.11·3-s + 1/32·4-s + 1.45·7-s + 0.234·9-s − 0.0347·12-s − 1.80·13-s − 0.999·16-s + 0.290·19-s − 1.61·21-s + 0.850·27-s + 0.0455·28-s − 0.761·31-s + 0.00733·36-s − 1.42·37-s + 2.00·39-s + 2.06·43-s + 1.11·48-s − 0.406·49-s − 0.0563·52-s − 0.322·57-s + 0.108·61-s + 0.341·63-s − 0.0624·64-s − 0.225·67-s − 1.94·73-s + 0.00908·76-s − 1.18·79-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5625 ^{s/2} \, \Gamma_{\C}(s+4)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(5625\)    =    \(3^{2} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(933.509\)
Root analytic conductor: \(5.52751\)
Motivic weight: \(8\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{75} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 5625,\ (\ :4, 4),\ 1)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(0.4967860339\)
\(L(\frac12)\) \(\approx\) \(0.4967860339\)
\(L(5)\) 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)$
bad3$C_2$ \( 1 + 10 p^{2} T + p^{8} T^{2} \)
5 \( 1 \)
good2$C_2^2$ \( 1 - p^{3} T^{2} + p^{16} T^{4} \)
7$C_2$ \( ( 1 - 250 p T + p^{8} T^{2} )^{2} \)
11$C_2^2$ \( 1 - 380283362 T^{2} + p^{16} T^{4} \)
13$C_2$ \( ( 1 + 25730 T + p^{8} T^{2} )^{2} \)
17$C_2^2$ \( 1 - 8342551298 T^{2} + p^{16} T^{4} \)
19$C_2$ \( ( 1 - 18938 T + p^{8} T^{2} )^{2} \)
23$C_2^2$ \( 1 + 64711613182 T^{2} + p^{16} T^{4} \)
29$C_2^2$ \( 1 - 788066452322 T^{2} + p^{16} T^{4} \)
31$C_2$ \( ( 1 + 11338 p T + p^{8} T^{2} )^{2} \)
37$C_2$ \( ( 1 + 1335170 T + p^{8} T^{2} )^{2} \)
41$C_2^2$ \( 1 - 12452468931842 T^{2} + p^{16} T^{4} \)
43$C_2$ \( ( 1 - 3526150 T + p^{8} T^{2} )^{2} \)
47$C_2^2$ \( 1 - 30967680304898 T^{2} + p^{16} T^{4} \)
53$C_2^2$ \( 1 - 80936075395298 T^{2} + p^{16} T^{4} \)
59$C_2^2$ \( 1 - 105562517046242 T^{2} + p^{16} T^{4} \)
61$C_2$ \( ( 1 - 753602 T + p^{8} T^{2} )^{2} \)
67$C_2$ \( ( 1 + 2268890 T + p^{8} T^{2} )^{2} \)
71$C_2^2$ \( 1 - 1001758688017922 T^{2} + p^{16} T^{4} \)
73$C_2$ \( ( 1 + 27672770 T + p^{8} T^{2} )^{2} \)
79$C_2$ \( ( 1 + 22980982 T + p^{8} T^{2} )^{2} \)
83$C_2^2$ \( 1 - 2352070843223138 T^{2} + p^{16} T^{4} \)
89$C_2^2$ \( 1 - 2600204109557762 T^{2} + p^{16} T^{4} \)
97$C_2$ \( ( 1 + 147271010 T + p^{8} T^{2} )^{2} \)
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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

−12.87612685381098833680385724101, −12.52339831250955989265689714946, −11.73431469155489004038660274857, −11.69757800541747014829048492897, −11.00236211332466004255926181015, −10.60147295225520905143866702319, −9.917739891777601320788017082330, −9.199949216526917596029386024782, −8.624654265334267569770350615686, −7.82391932586227961826660662715, −7.26397622977707208402442583836, −6.80378412140734767112081217749, −5.78749089268642832044809250103, −5.29539665397744324323830716469, −4.73450197968475733688084905638, −4.27872858250621541514994820717, −2.89974313146140004823428344933, −2.10203239877848330549034157547, −1.33365843388245525136650483603, −0.24164688123621780406836533867, 0.24164688123621780406836533867, 1.33365843388245525136650483603, 2.10203239877848330549034157547, 2.89974313146140004823428344933, 4.27872858250621541514994820717, 4.73450197968475733688084905638, 5.29539665397744324323830716469, 5.78749089268642832044809250103, 6.80378412140734767112081217749, 7.26397622977707208402442583836, 7.82391932586227961826660662715, 8.624654265334267569770350615686, 9.199949216526917596029386024782, 9.917739891777601320788017082330, 10.60147295225520905143866702319, 11.00236211332466004255926181015, 11.69757800541747014829048492897, 11.73431469155489004038660274857, 12.52339831250955989265689714946, 12.87612685381098833680385724101

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