Properties

Label 4-3850e2-1.1-c1e2-0-9
Degree $4$
Conductor $14822500$
Sign $1$
Analytic cond. $945.095$
Root an. cond. $5.54458$
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  − 4-s + 2·9-s + 2·11-s + 16-s − 4·29-s − 4·31-s − 2·36-s + 16·41-s − 2·44-s − 49-s + 20·59-s − 8·61-s − 64-s − 8·71-s + 32·79-s − 5·81-s + 12·89-s + 4·99-s + 12·109-s + 4·116-s + 3·121-s + 4·124-s + 127-s + 131-s + 137-s + 139-s + 2·144-s + ⋯
L(s)  = 1  − 1/2·4-s + 2/3·9-s + 0.603·11-s + 1/4·16-s − 0.742·29-s − 0.718·31-s − 1/3·36-s + 2.49·41-s − 0.301·44-s − 1/7·49-s + 2.60·59-s − 1.02·61-s − 1/8·64-s − 0.949·71-s + 3.60·79-s − 5/9·81-s + 1.27·89-s + 0.402·99-s + 1.14·109-s + 0.371·116-s + 3/11·121-s + 0.359·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 1/6·144-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 14822500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14822500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(14822500\)    =    \(2^{2} \cdot 5^{4} \cdot 7^{2} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(945.095\)
Root analytic conductor: \(5.54458\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{3850} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 14822500,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.520974034\)
\(L(\frac12)\) \(\approx\) \(2.520974034\)
\(L(\frac{3}{2})\) 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)$
bad2$C_2$ \( 1 + T^{2} \)
5 \( 1 \)
7$C_2$ \( 1 + T^{2} \)
11$C_1$ \( ( 1 - T )^{2} \)
good3$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - p T^{2} )^{2} \)
17$C_2$ \( ( 1 - p T^{2} )^{2} \)
19$C_2$ \( ( 1 + p T^{2} )^{2} \)
23$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
37$C_2^2$ \( 1 - 38 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 + 58 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 58 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 16 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - p T^{2} )^{2} \)
89$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
97$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
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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

−9.102851016375685355449668421369, −8.249730638991834068677682197615, −7.965457524100276422249051363171, −7.46968820428559218939756119381, −7.31002701081667890797758403167, −6.94414321189113682566220335605, −6.27522855776829928288571794061, −6.22503240157641070425265637778, −5.65592281802215143797257099778, −5.34439405294479514502170961882, −4.71318187376635366427411527391, −4.64201960507548573648465988986, −3.88833155141469657077173692272, −3.81766246686068104111632138353, −3.42073639932996228730471607585, −2.62971618310070572755495447571, −2.26480245974973972698210904445, −1.69134497070612708169180493123, −1.06917646227446723916463943455, −0.53891349391005806364379110360, 0.53891349391005806364379110360, 1.06917646227446723916463943455, 1.69134497070612708169180493123, 2.26480245974973972698210904445, 2.62971618310070572755495447571, 3.42073639932996228730471607585, 3.81766246686068104111632138353, 3.88833155141469657077173692272, 4.64201960507548573648465988986, 4.71318187376635366427411527391, 5.34439405294479514502170961882, 5.65592281802215143797257099778, 6.22503240157641070425265637778, 6.27522855776829928288571794061, 6.94414321189113682566220335605, 7.31002701081667890797758403167, 7.46968820428559218939756119381, 7.965457524100276422249051363171, 8.249730638991834068677682197615, 9.102851016375685355449668421369

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