Properties

Degree $4$
Conductor $3705625$
Sign $1$
Motivic weight $1$
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 3·4-s + 2·9-s + 2·11-s + 5·16-s + 12·29-s + 20·31-s + 6·36-s + 8·41-s + 6·44-s − 49-s − 4·59-s + 3·64-s − 24·71-s − 16·79-s − 5·81-s + 12·89-s + 4·99-s − 8·101-s + 28·109-s + 36·116-s + 3·121-s + 60·124-s + 127-s + 131-s + 137-s + 139-s + 10·144-s + ⋯
L(s)  = 1  + 3/2·4-s + 2/3·9-s + 0.603·11-s + 5/4·16-s + 2.22·29-s + 3.59·31-s + 36-s + 1.24·41-s + 0.904·44-s − 1/7·49-s − 0.520·59-s + 3/8·64-s − 2.84·71-s − 1.80·79-s − 5/9·81-s + 1.27·89-s + 0.402·99-s − 0.796·101-s + 2.68·109-s + 3.34·116-s + 3/11·121-s + 5.38·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 5/6·144-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3705625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3705625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(3705625\)    =    \(5^{4} \cdot 7^{2} \cdot 11^{2}\)
Sign: $1$
Motivic weight: \(1\)
Character: induced by $\chi_{1925} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 3705625,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(5.136299745\)
\(L(\frac12)\) \(\approx\) \(5.136299745\)
\(L(\frac{3}{2})\) 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)$
bad5 \( 1 \)
7$C_2$ \( 1 + T^{2} \)
11$C_1$ \( ( 1 - T )^{2} \)
good2$C_2^2$ \( 1 - 3 T^{2} + p^{2} T^{4} \)
3$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
17$C_2^2$ \( 1 - 18 T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 + p T^{2} )^{2} \)
23$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
37$C_2^2$ \( 1 - 38 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 + 58 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + p T^{2} )^{2} \)
67$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 82 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - p T^{2} )^{2} \)
89$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
97$C_2^2$ \( 1 - 94 T^{2} + p^{2} T^{4} \)
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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

−9.536122300748006273467764064858, −8.836199784352373422992317708260, −8.620955911609377922595486244277, −8.174492283481955528778622374035, −7.77640365622775922741520520700, −7.36584827890653499760017117079, −6.96291241772005650928043398338, −6.66106332374348458884267154469, −6.22609583639507254366492295648, −6.08293364216541482051082461888, −5.58462429807133170957624296101, −4.64462093272092201867296886648, −4.50297198453242379290658397893, −4.29041873153501702364939041195, −3.21988218125237988527319976309, −3.03178462562410694041578916247, −2.57530640345614782397191181207, −2.01156619727075697395212295152, −1.23714716037308604460338377373, −0.947003665188110625140996323384, 0.947003665188110625140996323384, 1.23714716037308604460338377373, 2.01156619727075697395212295152, 2.57530640345614782397191181207, 3.03178462562410694041578916247, 3.21988218125237988527319976309, 4.29041873153501702364939041195, 4.50297198453242379290658397893, 4.64462093272092201867296886648, 5.58462429807133170957624296101, 6.08293364216541482051082461888, 6.22609583639507254366492295648, 6.66106332374348458884267154469, 6.96291241772005650928043398338, 7.36584827890653499760017117079, 7.77640365622775922741520520700, 8.174492283481955528778622374035, 8.620955911609377922595486244277, 8.836199784352373422992317708260, 9.536122300748006273467764064858

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