Properties

Label 4-275e2-1.1-c1e2-0-10
Degree $4$
Conductor $75625$
Sign $-1$
Analytic cond. $4.82191$
Root an. cond. $1.48185$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4-s − 2·9-s − 3·16-s − 4·29-s − 8·31-s + 2·36-s − 4·41-s − 2·49-s + 4·61-s + 7·64-s − 16·71-s − 8·79-s − 5·81-s − 4·89-s − 4·101-s − 12·109-s + 4·116-s + 121-s + 8·124-s + 127-s + 131-s + 137-s + 139-s + 6·144-s + 149-s + 151-s + 157-s + ⋯
L(s)  = 1  − 1/2·4-s − 2/3·9-s − 3/4·16-s − 0.742·29-s − 1.43·31-s + 1/3·36-s − 0.624·41-s − 2/7·49-s + 0.512·61-s + 7/8·64-s − 1.89·71-s − 0.900·79-s − 5/9·81-s − 0.423·89-s − 0.398·101-s − 1.14·109-s + 0.371·116-s + 1/11·121-s + 0.718·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 1/2·144-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 75625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(75625\)    =    \(5^{4} \cdot 11^{2}\)
Sign: $-1$
Analytic conductor: \(4.82191\)
Root analytic conductor: \(1.48185\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 75625,\ (\ :1/2, 1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 \)
11$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
good2$C_2^2$ \( 1 + T^{2} + p^{2} T^{4} \)
3$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
7$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
13$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \)
17$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
23$C_2^2$ \( 1 - 38 T^{2} + p^{2} T^{4} \)
29$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
31$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \)
37$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
41$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
43$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 + 74 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
61$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
67$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \)
79$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \)
83$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \)
89$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
97$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
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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

−9.346749737675596470068677594467, −9.134258736602065766929881814924, −8.519257011027980098657609582100, −8.219611929124829730536393335027, −7.38918145196852137448307556962, −7.10234238875283960323976931212, −6.40213890318265051688157605702, −5.72262729885477183452585211358, −5.36882745968841857016892076672, −4.65831198775309809377533588381, −4.05922685946745316061925116204, −3.39161566630570559519822400727, −2.60874610245397079516983845116, −1.68042653722025704398000903398, 0, 1.68042653722025704398000903398, 2.60874610245397079516983845116, 3.39161566630570559519822400727, 4.05922685946745316061925116204, 4.65831198775309809377533588381, 5.36882745968841857016892076672, 5.72262729885477183452585211358, 6.40213890318265051688157605702, 7.10234238875283960323976931212, 7.38918145196852137448307556962, 8.219611929124829730536393335027, 8.519257011027980098657609582100, 9.134258736602065766929881814924, 9.346749737675596470068677594467

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