Properties

Label 4-5415e2-1.1-c1e2-0-1
Degree $4$
Conductor $29322225$
Sign $1$
Analytic cond. $1869.61$
Root an. cond. $6.57563$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s − 2·3-s + 4-s − 2·5-s + 4·6-s + 3·9-s + 4·10-s + 4·11-s − 2·12-s + 4·13-s + 4·15-s + 16-s + 8·17-s − 6·18-s − 2·20-s − 8·22-s + 4·23-s + 3·25-s − 8·26-s − 4·27-s − 8·30-s + 12·31-s + 2·32-s − 8·33-s − 16·34-s + 3·36-s + 4·37-s + ⋯
L(s)  = 1  − 1.41·2-s − 1.15·3-s + 1/2·4-s − 0.894·5-s + 1.63·6-s + 9-s + 1.26·10-s + 1.20·11-s − 0.577·12-s + 1.10·13-s + 1.03·15-s + 1/4·16-s + 1.94·17-s − 1.41·18-s − 0.447·20-s − 1.70·22-s + 0.834·23-s + 3/5·25-s − 1.56·26-s − 0.769·27-s − 1.46·30-s + 2.15·31-s + 0.353·32-s − 1.39·33-s − 2.74·34-s + 1/2·36-s + 0.657·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 29322225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29322225 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(29322225\)    =    \(3^{2} \cdot 5^{2} \cdot 19^{4}\)
Sign: $1$
Analytic conductor: \(1869.61\)
Root analytic conductor: \(6.57563\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{5415} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 29322225,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8593737908\)
\(L(\frac12)\) \(\approx\) \(0.8593737908\)
\(L(\frac{3}{2})\) 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)$
bad3$C_1$ \( ( 1 + T )^{2} \)
5$C_1$ \( ( 1 + T )^{2} \)
19 \( 1 \)
good2$D_{4}$ \( 1 + p T + 3 T^{2} + p^{2} T^{3} + p^{2} T^{4} \)
7$C_2^2$ \( 1 + 12 T^{2} + p^{2} T^{4} \)
11$C_2^2$ \( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
13$D_{4}$ \( 1 - 4 T + 28 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 - 8 T + 42 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
23$D_{4}$ \( 1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
29$C_2^2$ \( 1 + 56 T^{2} + p^{2} T^{4} \)
31$D_{4}$ \( 1 - 12 T + 90 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 - 4 T + 76 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 + 8 T + 80 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 - 16 T + 132 T^{2} - 16 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 + 4 T + 66 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
53$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
59$D_{4}$ \( 1 + 8 T + 62 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 + 8 T + 10 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 - 8 T + 118 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 - 16 T + 198 T^{2} - 16 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 + 4 T + 118 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
79$C_2$ \( ( 1 + p T^{2} )^{2} \)
83$D_{4}$ \( 1 - 20 T + 258 T^{2} - 20 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 - 8 T + 96 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 - 28 T + 372 T^{2} - 28 p T^{3} + p^{2} T^{4} \)
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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

−8.146882334194692924228764148594, −7.961297822617072408070681644727, −7.84721208835245750560075965943, −7.57994782542911195428636110027, −6.79235367677849492814735616609, −6.45922099321992712793353631351, −6.44410648581459075230236521785, −6.07512096931034695897690850642, −5.50432090076571703867907081640, −5.11376750850109856156349982156, −4.57090435077098425679869837570, −4.52638310859673068871195850233, −3.84098886181288094226346267997, −3.41621185245044703199838105347, −3.22185462012852144922909021079, −2.61154525792176734515021869087, −1.52098484718172889798692620556, −1.35762270765441350899846327149, −0.78409358858171443815299128567, −0.55654924884430229056701803874, 0.55654924884430229056701803874, 0.78409358858171443815299128567, 1.35762270765441350899846327149, 1.52098484718172889798692620556, 2.61154525792176734515021869087, 3.22185462012852144922909021079, 3.41621185245044703199838105347, 3.84098886181288094226346267997, 4.52638310859673068871195850233, 4.57090435077098425679869837570, 5.11376750850109856156349982156, 5.50432090076571703867907081640, 6.07512096931034695897690850642, 6.44410648581459075230236521785, 6.45922099321992712793353631351, 6.79235367677849492814735616609, 7.57994782542911195428636110027, 7.84721208835245750560075965943, 7.961297822617072408070681644727, 8.146882334194692924228764148594

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