Properties

Label 4-2925e2-1.1-c1e2-0-3
Degree $4$
Conductor $8555625$
Sign $1$
Analytic cond. $545.514$
Root an. cond. $4.83282$
Motivic weight $1$
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  − 10·11-s − 4·16-s + 12·19-s + 4·29-s + 4·31-s − 18·41-s + 13·49-s + 10·61-s − 18·71-s + 6·79-s + 22·89-s − 8·109-s + 53·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 169-s + 173-s + 40·176-s + 179-s + 181-s + ⋯
L(s)  = 1  − 3.01·11-s − 16-s + 2.75·19-s + 0.742·29-s + 0.718·31-s − 2.81·41-s + 13/7·49-s + 1.28·61-s − 2.13·71-s + 0.675·79-s + 2.33·89-s − 0.766·109-s + 4.81·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 0.0769·169-s + 0.0760·173-s + 3.01·176-s + 0.0747·179-s + 0.0743·181-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8555625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8555625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(8555625\)    =    \(3^{4} \cdot 5^{4} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(545.514\)
Root analytic conductor: \(4.83282\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 8555625,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.391150628\)
\(L(\frac12)\) \(\approx\) \(1.391150628\)
\(L(\frac{3}{2})\) 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)$
bad3 \( 1 \)
5 \( 1 \)
13$C_2$ \( 1 + T^{2} \)
good2$C_2$ \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \)
7$C_2^2$ \( 1 - 13 T^{2} + p^{2} T^{4} \)
11$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \)
17$C_2^2$ \( 1 + 15 T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 - 37 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
37$C_2^2$ \( 1 - 25 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 + 9 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 81 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 - 118 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + 9 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
79$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 150 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 11 T + p T^{2} )^{2} \)
97$C_2^2$ \( 1 - 73 T^{2} + p^{2} T^{4} \)
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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

−8.829263857119788647298578999098, −8.586103621581181900725593975737, −8.092596354693305673925359911071, −7.82487302143149079601235297046, −7.35204717627904115224397706269, −7.34346829919177236668793405576, −6.69535598486770611679807880819, −6.39637735725853034284312317121, −5.55212582984307859010543223628, −5.37876792662062960504528396697, −5.33833862112796163793126472740, −4.60104546697356809633934949749, −4.57761421465896162074233549631, −3.62232136451929325336670008464, −3.08186546067727107830046240922, −3.01918928634605029832731920160, −2.37046312943566954663974412878, −2.01034850016492351894221002289, −1.07835761842625531235173049725, −0.41274208485020410182178715212, 0.41274208485020410182178715212, 1.07835761842625531235173049725, 2.01034850016492351894221002289, 2.37046312943566954663974412878, 3.01918928634605029832731920160, 3.08186546067727107830046240922, 3.62232136451929325336670008464, 4.57761421465896162074233549631, 4.60104546697356809633934949749, 5.33833862112796163793126472740, 5.37876792662062960504528396697, 5.55212582984307859010543223628, 6.39637735725853034284312317121, 6.69535598486770611679807880819, 7.34346829919177236668793405576, 7.35204717627904115224397706269, 7.82487302143149079601235297046, 8.092596354693305673925359911071, 8.586103621581181900725593975737, 8.829263857119788647298578999098

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