Properties

Label 4-175e2-1.1-c1e2-0-6
Degree $4$
Conductor $30625$
Sign $-1$
Analytic cond. $1.95267$
Root an. cond. $1.18210$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $1$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 5·9-s − 6·11-s − 4·16-s − 10·29-s + 4·31-s + 4·41-s + 49-s + 20·59-s − 16·61-s − 16·71-s − 10·79-s + 16·81-s + 30·99-s + 24·101-s + 10·109-s + 5·121-s + 127-s + 131-s + 137-s + 139-s + 20·144-s + 149-s + 151-s + 157-s + 163-s + 167-s − 25·169-s + ⋯
L(s)  = 1  − 5/3·9-s − 1.80·11-s − 16-s − 1.85·29-s + 0.718·31-s + 0.624·41-s + 1/7·49-s + 2.60·59-s − 2.04·61-s − 1.89·71-s − 1.12·79-s + 16/9·81-s + 3.01·99-s + 2.38·101-s + 0.957·109-s + 5/11·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 5/3·144-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 1.92·169-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 30625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 30625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(30625\)    =    \(5^{4} \cdot 7^{2}\)
Sign: $-1$
Analytic conductor: \(1.95267\)
Root analytic conductor: \(1.18210\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 30625,\ (\ :1/2, 1/2),\ -1)\)

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 \( 1 \)
7$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
good2$C_2$ \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \)
3$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
11$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
17$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
19$C_2$ \( ( 1 + p T^{2} )^{2} \)
23$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
29$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
41$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
47$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
59$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
71$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
79$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
89$C_2$ \( ( 1 + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
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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

−10.28604192239361642576306157957, −9.734749354117359778149997929022, −9.098174271283233951402286060204, −8.569343753087953468815992872802, −8.274076028912949412078697949695, −7.38451703018051703490900915946, −7.30330020618146603033885123856, −5.99848698022791170310055133711, −5.93898740434763322465740513982, −5.14936161735373762545181523253, −4.63171310479474983768771180103, −3.57752582126466520607951640819, −2.76472422088648867371864611007, −2.25267400471040791170371820784, 0, 2.25267400471040791170371820784, 2.76472422088648867371864611007, 3.57752582126466520607951640819, 4.63171310479474983768771180103, 5.14936161735373762545181523253, 5.93898740434763322465740513982, 5.99848698022791170310055133711, 7.30330020618146603033885123856, 7.38451703018051703490900915946, 8.274076028912949412078697949695, 8.569343753087953468815992872802, 9.098174271283233951402286060204, 9.734749354117359778149997929022, 10.28604192239361642576306157957

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