Properties

Label 4-425e2-1.1-c3e2-0-7
Degree $4$
Conductor $180625$
Sign $1$
Analytic cond. $628.796$
Root an. cond. $5.00757$
Motivic weight $3$
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  + 7·4-s − 10·9-s − 48·11-s − 15·16-s − 232·19-s − 60·29-s − 344·31-s − 70·36-s − 684·41-s − 336·44-s − 98·49-s − 504·59-s + 220·61-s − 553·64-s − 1.41e3·71-s − 1.62e3·76-s + 968·79-s − 629·81-s + 1.54e3·89-s + 480·99-s − 420·101-s + 2.37e3·109-s − 420·116-s − 934·121-s − 2.40e3·124-s + 127-s + 131-s + ⋯
L(s)  = 1  + 7/8·4-s − 0.370·9-s − 1.31·11-s − 0.234·16-s − 2.80·19-s − 0.384·29-s − 1.99·31-s − 0.324·36-s − 2.60·41-s − 1.15·44-s − 2/7·49-s − 1.11·59-s + 0.461·61-s − 1.08·64-s − 2.36·71-s − 2.45·76-s + 1.37·79-s − 0.862·81-s + 1.84·89-s + 0.487·99-s − 0.413·101-s + 2.08·109-s − 0.336·116-s − 0.701·121-s − 1.74·124-s + 0.000698·127-s + 0.000666·131-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 180625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 180625 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(180625\)    =    \(5^{4} \cdot 17^{2}\)
Sign: $1$
Analytic conductor: \(628.796\)
Root analytic conductor: \(5.00757\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 180625,\ (\ :3/2, 3/2),\ 1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) 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 \( 1 \)
17$C_2$ \( 1 + p^{2} T^{2} \)
good2$C_2^2$ \( 1 - 7 T^{2} + p^{6} T^{4} \)
3$C_2^2$ \( 1 + 10 T^{2} + p^{6} T^{4} \)
7$C_2^2$ \( 1 + 2 p^{2} T^{2} + p^{6} T^{4} \)
11$C_2$ \( ( 1 + 24 T + p^{3} T^{2} )^{2} \)
13$C_2^2$ \( 1 - 1030 T^{2} + p^{6} T^{4} \)
19$C_2$ \( ( 1 + 116 T + p^{3} T^{2} )^{2} \)
23$C_2^2$ \( 1 - 20734 T^{2} + p^{6} T^{4} \)
29$C_2$ \( ( 1 + 30 T + p^{3} T^{2} )^{2} \)
31$C_2$ \( ( 1 + 172 T + p^{3} T^{2} )^{2} \)
37$C_2^2$ \( 1 - 97942 T^{2} + p^{6} T^{4} \)
41$C_2$ \( ( 1 + 342 T + p^{3} T^{2} )^{2} \)
43$C_2^2$ \( 1 - 137110 T^{2} + p^{6} T^{4} \)
47$C_2^2$ \( 1 - 124702 T^{2} + p^{6} T^{4} \)
53$C_2^2$ \( 1 - 70 p^{2} T^{2} + p^{6} T^{4} \)
59$C_2$ \( ( 1 + 252 T + p^{3} T^{2} )^{2} \)
61$C_2$ \( ( 1 - 110 T + p^{3} T^{2} )^{2} \)
67$C_2^2$ \( 1 - 367270 T^{2} + p^{6} T^{4} \)
71$C_2$ \( ( 1 + 708 T + p^{3} T^{2} )^{2} \)
73$C_2^2$ \( 1 - 646990 T^{2} + p^{6} T^{4} \)
79$C_2$ \( ( 1 - 484 T + p^{3} T^{2} )^{2} \)
83$C_2^2$ \( 1 - 572038 T^{2} + p^{6} T^{4} \)
89$C_2$ \( ( 1 - 774 T + p^{3} T^{2} )^{2} \)
97$C_2^2$ \( 1 - 1679422 T^{2} + p^{6} T^{4} \)
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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

−10.48262812718079295191751081100, −10.45003594599013903395813572461, −9.748953406935180400184091027290, −8.916466472894863703065734888805, −8.772156140594064990213604724660, −8.317040883954338335083503122259, −7.52430748825825200745536874118, −7.48714667875867588271705789486, −6.61879848170905525825960160666, −6.41697704808275461280133467235, −5.84798190277569519804114226122, −5.21524494767023274350368859995, −4.76542088483137001492988901804, −4.06550794064444127682719866466, −3.37760450585846030072698215390, −2.73931685831830878603720113640, −2.00323208346037421144177330888, −1.81424147318915866618309580227, 0, 0, 1.81424147318915866618309580227, 2.00323208346037421144177330888, 2.73931685831830878603720113640, 3.37760450585846030072698215390, 4.06550794064444127682719866466, 4.76542088483137001492988901804, 5.21524494767023274350368859995, 5.84798190277569519804114226122, 6.41697704808275461280133467235, 6.61879848170905525825960160666, 7.48714667875867588271705789486, 7.52430748825825200745536874118, 8.317040883954338335083503122259, 8.772156140594064990213604724660, 8.916466472894863703065734888805, 9.748953406935180400184091027290, 10.45003594599013903395813572461, 10.48262812718079295191751081100

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