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Properties

Label 4-175e2-1.1-c0e2-0-0
Degree $4$
Conductor $30625$
Sign $1$
Analytic cond. $0.00762764$
Root an. cond. $0.295527$
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  + 4-s − 2·9-s − 2·11-s + 2·29-s − 2·36-s − 2·44-s − 49-s − 64-s − 2·71-s + 2·79-s + 3·81-s + 4·99-s + 2·109-s + 2·116-s + 121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 2·169-s + 173-s + 179-s + ⋯
L(s)  = 1  + 4-s − 2·9-s − 2·11-s + 2·29-s − 2·36-s − 2·44-s − 49-s − 64-s − 2·71-s + 2·79-s + 3·81-s + 4·99-s + 2·109-s + 2·116-s + 121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 2·169-s + 173-s + 179-s + ⋯

Functional equation

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

Invariants

Degree: \(4\)
Conductor: \(30625\)    =    \(5^{4} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(0.00762764\)
Root analytic conductor: \(0.295527\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 30625,\ (\ :0, 0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4646724337\)
\(L(\frac12)\) \(\approx\) \(0.4646724337\)
\(L(1)\) 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 \( 1 \)
7$C_2$ \( 1 + T^{2} \)
good2$C_2^2$ \( 1 - T^{2} + T^{4} \)
3$C_2$ \( ( 1 + T^{2} )^{2} \)
11$C_2$ \( ( 1 + T + T^{2} )^{2} \)
13$C_2$ \( ( 1 + T^{2} )^{2} \)
17$C_2$ \( ( 1 + T^{2} )^{2} \)
19$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
23$C_2^2$ \( 1 - T^{2} + T^{4} \)
29$C_2$ \( ( 1 - T + T^{2} )^{2} \)
31$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
37$C_2^2$ \( 1 - T^{2} + T^{4} \)
41$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
43$C_2^2$ \( 1 - T^{2} + T^{4} \)
47$C_2$ \( ( 1 + T^{2} )^{2} \)
53$C_2$ \( ( 1 + T^{2} )^{2} \)
59$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
61$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
67$C_2^2$ \( 1 - T^{2} + T^{4} \)
71$C_2$ \( ( 1 + T + T^{2} )^{2} \)
73$C_2$ \( ( 1 + T^{2} )^{2} \)
79$C_2$ \( ( 1 - T + T^{2} )^{2} \)
83$C_2$ \( ( 1 + T^{2} )^{2} \)
89$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
97$C_2$ \( ( 1 + T^{2} )^{2} \)
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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

−13.51079718904401974351850817746, −12.53910350627727582540120298789, −12.14444489939289199289128893931, −11.70372483322166441461992033855, −11.03055908768080276870833170769, −11.01720550911740402855652502622, −10.29555842557836072172498395314, −9.967080819986157002800541397073, −8.948939647607505608978697134519, −8.660735445337753490462854231169, −7.892558870453853779324276599677, −7.82270203266143885773618920167, −6.90392282848403870644246039249, −6.28445743743243721438375152643, −5.86088782635053946864062341182, −5.16453359959286448991575163680, −4.70197917174654042535704225276, −3.27454930028603520033152893804, −2.78551367007755798973085985436, −2.28332171512070147804322165723, 2.28332171512070147804322165723, 2.78551367007755798973085985436, 3.27454930028603520033152893804, 4.70197917174654042535704225276, 5.16453359959286448991575163680, 5.86088782635053946864062341182, 6.28445743743243721438375152643, 6.90392282848403870644246039249, 7.82270203266143885773618920167, 7.892558870453853779324276599677, 8.660735445337753490462854231169, 8.948939647607505608978697134519, 9.967080819986157002800541397073, 10.29555842557836072172498395314, 11.01720550911740402855652502622, 11.03055908768080276870833170769, 11.70372483322166441461992033855, 12.14444489939289199289128893931, 12.53910350627727582540120298789, 13.51079718904401974351850817746

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