Properties

Label 4-5e4-1.1-c3e2-0-1
Degree $4$
Conductor $625$
Sign $1$
Analytic cond. $2.17576$
Root an. cond. $1.21451$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 15·4-s + 5·9-s − 86·11-s + 161·16-s + 70·19-s − 320·29-s + 84·31-s + 75·36-s − 406·41-s − 1.29e3·44-s + 650·49-s + 560·59-s − 1.03e3·61-s + 1.45e3·64-s + 824·71-s + 1.05e3·76-s − 1.02e3·79-s − 704·81-s + 1.89e3·89-s − 430·99-s + 2.60e3·101-s − 2.14e3·109-s − 4.80e3·116-s + 2.88e3·121-s + 1.26e3·124-s + 127-s + 131-s + ⋯
L(s)  = 1  + 15/8·4-s + 5/27·9-s − 2.35·11-s + 2.51·16-s + 0.845·19-s − 2.04·29-s + 0.486·31-s + 0.347·36-s − 1.54·41-s − 4.41·44-s + 1.89·49-s + 1.23·59-s − 2.17·61-s + 2.84·64-s + 1.37·71-s + 1.58·76-s − 1.45·79-s − 0.965·81-s + 2.25·89-s − 0.436·99-s + 2.56·101-s − 1.88·109-s − 3.84·116-s + 2.16·121-s + 0.912·124-s + 0.000698·127-s + 0.000666·131-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 625 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(625\)    =    \(5^{4}\)
Sign: $1$
Analytic conductor: \(2.17576\)
Root analytic conductor: \(1.21451\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 625,\ (\ :3/2, 3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(1.669098600\)
\(L(\frac12)\) \(\approx\) \(1.669098600\)
\(L(\frac{5}{2})\) 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)$
bad5 \( 1 \)
good2$C_2^2$ \( 1 - 15 T^{2} + p^{6} T^{4} \)
3$C_2^2$ \( 1 - 5 T^{2} + p^{6} T^{4} \)
7$C_2^2$ \( 1 - 650 T^{2} + p^{6} T^{4} \)
11$C_2$ \( ( 1 + 43 T + p^{3} T^{2} )^{2} \)
13$C_2^2$ \( 1 - 3610 T^{2} + p^{6} T^{4} \)
17$C_2^2$ \( 1 - 1545 T^{2} + p^{6} T^{4} \)
19$C_2$ \( ( 1 - 35 T + p^{3} T^{2} )^{2} \)
23$C_2^2$ \( 1 + 1910 T^{2} + p^{6} T^{4} \)
29$C_2$ \( ( 1 + 160 T + p^{3} T^{2} )^{2} \)
31$C_2$ \( ( 1 - 42 T + p^{3} T^{2} )^{2} \)
37$C_2^2$ \( 1 - 2710 T^{2} + p^{6} T^{4} \)
41$C_2$ \( ( 1 + 203 T + p^{3} T^{2} )^{2} \)
43$C_2^2$ \( 1 - 150550 T^{2} + p^{6} T^{4} \)
47$C_2^2$ \( 1 - 169230 T^{2} + p^{6} T^{4} \)
53$C_2^2$ \( 1 - 291030 T^{2} + p^{6} T^{4} \)
59$C_2$ \( ( 1 - 280 T + p^{3} T^{2} )^{2} \)
61$C_2$ \( ( 1 + 518 T + p^{3} T^{2} )^{2} \)
67$C_2^2$ \( 1 - 581645 T^{2} + p^{6} T^{4} \)
71$C_2$ \( ( 1 - 412 T + p^{3} T^{2} )^{2} \)
73$C_2^2$ \( 1 - 195865 T^{2} + p^{6} T^{4} \)
79$C_2$ \( ( 1 + 510 T + p^{3} T^{2} )^{2} \)
83$C_2^2$ \( 1 - 539845 T^{2} + p^{6} T^{4} \)
89$C_2$ \( ( 1 - 945 T + p^{3} T^{2} )^{2} \)
97$C_2^2$ \( 1 - 272830 T^{2} + p^{6} T^{4} \)
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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

−17.05076349983359168007171014969, −16.81118026148155788098881075705, −15.80886096184293155183808000568, −15.77609858885254778552409524511, −15.24155034372754449354346155234, −14.53758059600749516423429574126, −13.39915190482900629182656295589, −13.04448246995962569555365058116, −12.11238564977815600494094980373, −11.64019346937698360220785054931, −10.69474654262369930534458398258, −10.55698726784516620872082359541, −9.669932974988978783458582004562, −8.280199123235320798710994728855, −7.51585598831460242060290849714, −7.16710040603866174443234661203, −5.91497291920024704164320052484, −5.25846042110859208081852434226, −3.22568055556344840104013981859, −2.17753244888952821685534721355, 2.17753244888952821685534721355, 3.22568055556344840104013981859, 5.25846042110859208081852434226, 5.91497291920024704164320052484, 7.16710040603866174443234661203, 7.51585598831460242060290849714, 8.280199123235320798710994728855, 9.669932974988978783458582004562, 10.55698726784516620872082359541, 10.69474654262369930534458398258, 11.64019346937698360220785054931, 12.11238564977815600494094980373, 13.04448246995962569555365058116, 13.39915190482900629182656295589, 14.53758059600749516423429574126, 15.24155034372754449354346155234, 15.77609858885254778552409524511, 15.80886096184293155183808000568, 16.81118026148155788098881075705, 17.05076349983359168007171014969

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