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Properties

Label 4-115e2-1.1-c1e2-0-2
Degree $4$
Conductor $13225$
Sign $1$
Analytic cond. $0.843237$
Root an. cond. $0.958269$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·2-s − 2·3-s + 4·4-s − 2·5-s + 6·6-s − 2·7-s − 3·8-s − 3·9-s + 6·10-s − 2·11-s − 8·12-s − 8·13-s + 6·14-s + 4·15-s + 3·16-s − 4·17-s + 9·18-s + 2·19-s − 8·20-s + 4·21-s + 6·22-s − 2·23-s + 6·24-s + 3·25-s + 24·26-s + 14·27-s − 8·28-s + ⋯
L(s)  = 1  − 2.12·2-s − 1.15·3-s + 2·4-s − 0.894·5-s + 2.44·6-s − 0.755·7-s − 1.06·8-s − 9-s + 1.89·10-s − 0.603·11-s − 2.30·12-s − 2.21·13-s + 1.60·14-s + 1.03·15-s + 3/4·16-s − 0.970·17-s + 2.12·18-s + 0.458·19-s − 1.78·20-s + 0.872·21-s + 1.27·22-s − 0.417·23-s + 1.22·24-s + 3/5·25-s + 4.70·26-s + 2.69·27-s − 1.51·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 13225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13225 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(13225\)    =    \(5^{2} \cdot 23^{2}\)
Sign: $1$
Analytic conductor: \(0.843237\)
Root analytic conductor: \(0.958269\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 13225,\ (\ :1/2, 1/2),\ 1)\)

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$C_1$ \( ( 1 + T )^{2} \)
23$C_1$ \( ( 1 + T )^{2} \)
good2$C_2^2$ \( 1 + 3 T + 5 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
3$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
7$D_{4}$ \( 1 + 2 T + 10 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
11$D_{4}$ \( 1 + 2 T + 18 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
13$D_{4}$ \( 1 + 8 T + 37 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 + 4 T + 18 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
19$C_4$ \( 1 - 2 T - 6 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 + 10 T + 63 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 - 4 T + 61 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 + 6 T + 38 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 + 6 T + 71 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 + 6 T + 50 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 - 10 T + 99 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
53$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
59$C_2^2$ \( 1 + 38 T^{2} + p^{2} T^{4} \)
61$D_{4}$ \( 1 - 2 T - 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 - 6 T + 98 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 + 8 T + 153 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
73$C_2^2$ \( 1 + 101 T^{2} + p^{2} T^{4} \)
79$D_{4}$ \( 1 - 22 T + 274 T^{2} - 22 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 + 4 T + 150 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 - 10 T + 198 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 - 10 T + 94 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
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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

−13.06088547824854618800549466457, −12.41299076453725953754550936621, −12.00967254881949020851948198426, −11.68470866415916587033008191573, −11.00402456370653763196805639625, −10.68569453507544161612059950237, −10.01118667745436500401957574714, −9.649899907776049668127822314988, −9.024219188132851231160899094552, −8.603493233501857420483830647330, −7.86136208353759043663879641278, −7.58119184561242268186748378465, −6.80983195663352852256465199092, −6.23856264103050152148309133917, −5.24824667529767515960464977819, −4.92675119180659763498456396168, −3.43173593624090948026502900929, −2.46890853303187974438036531558, 0, 0, 2.46890853303187974438036531558, 3.43173593624090948026502900929, 4.92675119180659763498456396168, 5.24824667529767515960464977819, 6.23856264103050152148309133917, 6.80983195663352852256465199092, 7.58119184561242268186748378465, 7.86136208353759043663879641278, 8.603493233501857420483830647330, 9.024219188132851231160899094552, 9.649899907776049668127822314988, 10.01118667745436500401957574714, 10.68569453507544161612059950237, 11.00402456370653763196805639625, 11.68470866415916587033008191573, 12.00967254881949020851948198426, 12.41299076453725953754550936621, 13.06088547824854618800549466457

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