Properties

Label 4-2475e2-1.1-c1e2-0-17
Degree $4$
Conductor $6125625$
Sign $1$
Analytic cond. $390.575$
Root an. cond. $4.44555$
Motivic weight $1$
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  − 2-s + 5·7-s − 8-s + 2·11-s + 10·13-s − 5·14-s − 16-s − 3·17-s − 2·19-s − 2·22-s + 11·23-s − 10·26-s + 9·29-s + 6·31-s + 6·32-s + 3·34-s + 12·37-s + 2·38-s + 4·41-s − 11·46-s + 6·47-s + 8·49-s − 53-s − 5·56-s − 9·58-s + 14·59-s − 5·61-s + ⋯
L(s)  = 1  − 0.707·2-s + 1.88·7-s − 0.353·8-s + 0.603·11-s + 2.77·13-s − 1.33·14-s − 1/4·16-s − 0.727·17-s − 0.458·19-s − 0.426·22-s + 2.29·23-s − 1.96·26-s + 1.67·29-s + 1.07·31-s + 1.06·32-s + 0.514·34-s + 1.97·37-s + 0.324·38-s + 0.624·41-s − 1.62·46-s + 0.875·47-s + 8/7·49-s − 0.137·53-s − 0.668·56-s − 1.18·58-s + 1.82·59-s − 0.640·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6125625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6125625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(6125625\)    =    \(3^{4} \cdot 5^{4} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(390.575\)
Root analytic conductor: \(4.44555\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{2475} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 6125625,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.206757715\)
\(L(\frac12)\) \(\approx\) \(3.206757715\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 \)
11$C_1$ \( ( 1 - T )^{2} \)
good2$D_{4}$ \( 1 + T + T^{2} + p T^{3} + p^{2} T^{4} \)
7$D_{4}$ \( 1 - 5 T + 17 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
17$D_{4}$ \( 1 + 3 T + 7 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
19$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
23$D_{4}$ \( 1 - 11 T + 73 T^{2} - 11 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 - 9 T + 49 T^{2} - 9 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 - 6 T + 19 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 - 12 T + 97 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 - 4 T + 73 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
43$C_2^2$ \( 1 + 34 T^{2} + p^{2} T^{4} \)
47$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
53$D_{4}$ \( 1 + T + 103 T^{2} + p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 - 14 T + 115 T^{2} - 14 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 + 5 T + 99 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
67$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
71$D_{4}$ \( 1 + 2 T + 130 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 + 5 T + 123 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 + 11 T + 159 T^{2} + 11 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 + 11 T + 115 T^{2} + 11 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 + 7 T + 187 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 - 27 T + 373 T^{2} - 27 p T^{3} + p^{2} T^{4} \)
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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

−8.779940527582194637969647508480, −8.700451983004077040091605914921, −8.452648218480890210117134528054, −8.328005907094684800681941219256, −7.77052298978826926256323429642, −7.17780380781200135307456301813, −6.93028653700106942772943223414, −6.33876613364021270731345753834, −6.07991306065274212854240727357, −5.86338720621865247478918543228, −5.05452977537316869129938134035, −4.68890903087023092055822135955, −4.34117290240050617162833815136, −4.11494686667864349522574341279, −3.36549627643610455398469836119, −2.78632753050404375283747449689, −2.42897203348670481463334549873, −1.52662039952261864760430324013, −1.05921578965706264750828902003, −0.928841580913922005754582762206, 0.928841580913922005754582762206, 1.05921578965706264750828902003, 1.52662039952261864760430324013, 2.42897203348670481463334549873, 2.78632753050404375283747449689, 3.36549627643610455398469836119, 4.11494686667864349522574341279, 4.34117290240050617162833815136, 4.68890903087023092055822135955, 5.05452977537316869129938134035, 5.86338720621865247478918543228, 6.07991306065274212854240727357, 6.33876613364021270731345753834, 6.93028653700106942772943223414, 7.17780380781200135307456301813, 7.77052298978826926256323429642, 8.328005907094684800681941219256, 8.452648218480890210117134528054, 8.700451983004077040091605914921, 8.779940527582194637969647508480

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