Properties

Label 4-175e2-1.1-c4e2-0-0
Degree $4$
Conductor $30625$
Sign $1$
Analytic cond. $327.238$
Root an. cond. $4.25320$
Motivic weight $4$
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  − 32·4-s − 127·9-s − 146·11-s + 768·16-s + 2.30e3·29-s + 4.06e3·36-s + 4.67e3·44-s − 2.40e3·49-s − 1.63e4·64-s − 2.01e4·71-s − 2.43e4·79-s + 9.56e3·81-s + 1.85e4·99-s + 2.87e4·109-s − 7.37e4·116-s − 1.32e4·121-s + 127-s + 131-s + 137-s + 139-s − 9.75e4·144-s + 149-s + 151-s + 157-s + 163-s + 167-s + 5.65e4·169-s + ⋯
L(s)  = 1  − 2·4-s − 1.56·9-s − 1.20·11-s + 3·16-s + 2.74·29-s + 3.13·36-s + 2.41·44-s − 49-s − 4·64-s − 3.99·71-s − 3.89·79-s + 1.45·81-s + 1.89·99-s + 2.41·109-s − 5.48·116-s − 0.908·121-s + 6.20e−5·127-s + 5.82e−5·131-s + 5.32e−5·137-s + 5.17e−5·139-s − 4.70·144-s + 4.50e−5·149-s + 4.38e−5·151-s + 4.05e−5·157-s + 3.76e−5·163-s + 3.58e−5·167-s + 1.98·169-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 30625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 30625 ^{s/2} \, \Gamma_{\C}(s+2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(30625\)    =    \(5^{4} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(327.238\)
Root analytic conductor: \(4.25320\)
Motivic weight: \(4\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 30625,\ (\ :2, 2),\ 1)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.1811732437\)
\(L(\frac12)\) \(\approx\) \(0.1811732437\)
\(L(3)\) 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 \( 1 \)
7$C_2$ \( 1 + p^{4} T^{2} \)
good2$C_2$ \( ( 1 + p^{4} T^{2} )^{2} \)
3$C_2^2$ \( 1 + 127 T^{2} + p^{8} T^{4} \)
11$C_2$ \( ( 1 + 73 T + p^{4} T^{2} )^{2} \)
13$C_2^2$ \( 1 - 56593 T^{2} + p^{8} T^{4} \)
17$C_2^2$ \( 1 - 97873 T^{2} + p^{8} T^{4} \)
19$C_1$$\times$$C_1$ \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \)
23$C_2$ \( ( 1 + p^{4} T^{2} )^{2} \)
29$C_2$ \( ( 1 - 1153 T + p^{4} T^{2} )^{2} \)
31$C_1$$\times$$C_1$ \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \)
37$C_2$ \( ( 1 + p^{4} T^{2} )^{2} \)
41$C_1$$\times$$C_1$ \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \)
43$C_2$ \( ( 1 + p^{4} T^{2} )^{2} \)
47$C_2^2$ \( 1 + 2191487 T^{2} + p^{8} T^{4} \)
53$C_2$ \( ( 1 + p^{4} T^{2} )^{2} \)
59$C_1$$\times$$C_1$ \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \)
61$C_1$$\times$$C_1$ \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \)
67$C_2$ \( ( 1 + p^{4} T^{2} )^{2} \)
71$C_2$ \( ( 1 + 10078 T + p^{4} T^{2} )^{2} \)
73$C_2^2$ \( 1 + 33491522 T^{2} + p^{8} T^{4} \)
79$C_2$ \( ( 1 + 12167 T + p^{4} T^{2} )^{2} \)
83$C_2^2$ \( 1 - 54186718 T^{2} + p^{8} T^{4} \)
89$C_1$$\times$$C_1$ \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \)
97$C_2^2$ \( 1 - 165613873 T^{2} + p^{8} T^{4} \)
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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

−12.74517407084250623819375037999, −11.61255330255000158007586559637, −11.58491129346635445740829955346, −10.39949357406696623372566339528, −10.32600946677662226251749735440, −9.882068079184456581102345115291, −8.956881689614059318660516929864, −8.783651952700924684966521810987, −8.357377339723924411560611051590, −7.933292905023312642315167536233, −7.27625325138586651737267955245, −6.11380649219034462579692243013, −5.84541391350072504502295340745, −5.08990311407056413090264769220, −4.75573542553034365660473373615, −4.12400070079137062066383297976, −2.95453367665585126965804070782, −2.93584000788115948774604698036, −1.20475748403600318982388483949, −0.18322272186938216413664157224, 0.18322272186938216413664157224, 1.20475748403600318982388483949, 2.93584000788115948774604698036, 2.95453367665585126965804070782, 4.12400070079137062066383297976, 4.75573542553034365660473373615, 5.08990311407056413090264769220, 5.84541391350072504502295340745, 6.11380649219034462579692243013, 7.27625325138586651737267955245, 7.933292905023312642315167536233, 8.357377339723924411560611051590, 8.783651952700924684966521810987, 8.956881689614059318660516929864, 9.882068079184456581102345115291, 10.32600946677662226251749735440, 10.39949357406696623372566339528, 11.58491129346635445740829955346, 11.61255330255000158007586559637, 12.74517407084250623819375037999

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