Properties

Label 4-2015e2-1.1-c0e2-0-1
Degree $4$
Conductor $4060225$
Sign $1$
Analytic cond. $1.01126$
Root an. cond. $1.00280$
Motivic weight $0$
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  + 2·3-s − 4-s + 2·5-s + 9-s − 2·12-s + 13-s + 4·15-s − 17-s + 19-s − 2·20-s − 23-s + 3·25-s − 2·27-s + 2·31-s − 36-s − 37-s + 2·39-s + 41-s − 43-s + 2·45-s − 49-s − 2·51-s − 52-s + 2·53-s + 2·57-s + 59-s − 4·60-s + ⋯
L(s)  = 1  + 2·3-s − 4-s + 2·5-s + 9-s − 2·12-s + 13-s + 4·15-s − 17-s + 19-s − 2·20-s − 23-s + 3·25-s − 2·27-s + 2·31-s − 36-s − 37-s + 2·39-s + 41-s − 43-s + 2·45-s − 49-s − 2·51-s − 52-s + 2·53-s + 2·57-s + 59-s − 4·60-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4060225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4060225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(4060225\)    =    \(5^{2} \cdot 13^{2} \cdot 31^{2}\)
Sign: $1$
Analytic conductor: \(1.01126\)
Root analytic conductor: \(1.00280\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 4060225,\ (\ :0, 0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.712141317\)
\(L(\frac12)\) \(\approx\) \(2.712141317\)
\(L(1)\) 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$C_1$ \( ( 1 - T )^{2} \)
13$C_2$ \( 1 - T + T^{2} \)
31$C_1$ \( ( 1 - T )^{2} \)
good2$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
3$C_2$ \( ( 1 - T + T^{2} )^{2} \)
7$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
11$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
17$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 - T + T^{2} ) \)
19$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T + T^{2} ) \)
23$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 - T + T^{2} ) \)
29$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
37$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 - T + T^{2} ) \)
41$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T + T^{2} ) \)
43$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 - T + T^{2} ) \)
47$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
53$C_2$ \( ( 1 - T + T^{2} )^{2} \)
59$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T + T^{2} ) \)
61$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
67$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
71$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T + T^{2} ) \)
73$C_1$ \( ( 1 + T )^{4} \)
79$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
83$C_2$ \( ( 1 - T + T^{2} )^{2} \)
89$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
97$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
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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

−9.564342273201083170771894671525, −8.973437830696771282244825353059, −8.835694282105224245604335065364, −8.446012274705330453190676880844, −8.433425281356616107840097949216, −7.73077641416160844032217067853, −7.38635798894064934213762770605, −6.67191979253310352585543787114, −6.33630492288886064595061644760, −6.07674539639486189808516648918, −5.36473189576526764261045781321, −5.27118353993412501623955610518, −4.66470846941597576815250329669, −3.94787282790723248587260014878, −3.83518184903051328584385722872, −3.03126444200120827247587212754, −2.73804189827363699147307946678, −2.33761764184055697666747542262, −1.82371456114366736857404340108, −1.18305123503357502726197551060, 1.18305123503357502726197551060, 1.82371456114366736857404340108, 2.33761764184055697666747542262, 2.73804189827363699147307946678, 3.03126444200120827247587212754, 3.83518184903051328584385722872, 3.94787282790723248587260014878, 4.66470846941597576815250329669, 5.27118353993412501623955610518, 5.36473189576526764261045781321, 6.07674539639486189808516648918, 6.33630492288886064595061644760, 6.67191979253310352585543787114, 7.38635798894064934213762770605, 7.73077641416160844032217067853, 8.433425281356616107840097949216, 8.446012274705330453190676880844, 8.835694282105224245604335065364, 8.973437830696771282244825353059, 9.564342273201083170771894671525

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