Properties

Label 4-1575e2-1.1-c1e2-0-9
Degree $4$
Conductor $2480625$
Sign $1$
Analytic cond. $158.166$
Root an. cond. $3.54632$
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  + 4·4-s + 6·11-s + 12·16-s − 4·19-s + 6·29-s − 8·31-s + 24·41-s + 24·44-s − 49-s + 16·61-s + 32·64-s − 16·76-s + 2·79-s − 24·89-s − 12·101-s + 14·109-s + 24·116-s + 5·121-s − 32·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + ⋯
L(s)  = 1  + 2·4-s + 1.80·11-s + 3·16-s − 0.917·19-s + 1.11·29-s − 1.43·31-s + 3.74·41-s + 3.61·44-s − 1/7·49-s + 2.04·61-s + 4·64-s − 1.83·76-s + 0.225·79-s − 2.54·89-s − 1.19·101-s + 1.34·109-s + 2.22·116-s + 5/11·121-s − 2.87·124-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 + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2480625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2480625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(2480625\)    =    \(3^{4} \cdot 5^{4} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(158.166\)
Root analytic conductor: \(3.54632\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1575} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 2480625,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(5.336109469\)
\(L(\frac12)\) \(\approx\) \(5.336109469\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7$C_2$ \( 1 + T^{2} \)
good2$C_2$ \( ( 1 - p T^{2} )^{2} \)
11$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
13$C_2^2$ \( 1 - T^{2} + p^{2} T^{4} \)
17$C_2^2$ \( 1 - 25 T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
41$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 13 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 + 38 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 - 118 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 142 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
97$C_2^2$ \( 1 - 193 T^{2} + p^{2} T^{4} \)
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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

−9.478365874869076816654096198730, −9.399349594446858084312927631376, −8.752835688899261381175208934640, −8.476032780663399742993589289873, −7.75858576815090926024565943630, −7.71096125778843593527190079586, −6.99040980534900117286456856642, −6.88226149251641680396258570888, −6.40995117947110185873739957064, −6.18613744789754897364081728834, −5.58416559008108437440761062076, −5.44082946768127261295685161203, −4.28902274945067524053534047938, −4.22280382744029398440466937073, −3.60758126763942390625852856145, −3.08984098876031870664575516273, −2.38979472144677827315004858676, −2.22062672683477142719393612124, −1.37921665283021032989857998998, −0.984811152605994644250568524101, 0.984811152605994644250568524101, 1.37921665283021032989857998998, 2.22062672683477142719393612124, 2.38979472144677827315004858676, 3.08984098876031870664575516273, 3.60758126763942390625852856145, 4.22280382744029398440466937073, 4.28902274945067524053534047938, 5.44082946768127261295685161203, 5.58416559008108437440761062076, 6.18613744789754897364081728834, 6.40995117947110185873739957064, 6.88226149251641680396258570888, 6.99040980534900117286456856642, 7.71096125778843593527190079586, 7.75858576815090926024565943630, 8.476032780663399742993589289873, 8.752835688899261381175208934640, 9.399349594446858084312927631376, 9.478365874869076816654096198730

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