Properties

Label 4-2475e2-1.1-c3e2-0-3
Degree $4$
Conductor $6125625$
Sign $1$
Analytic cond. $21324.6$
Root an. cond. $12.0842$
Motivic weight $3$
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-s − 11·4-s + 4·7-s + 15·8-s + 22·11-s + 90·13-s − 4·14-s + 61·16-s − 16·17-s − 170·19-s − 22·22-s − 124·23-s − 90·26-s − 44·28-s + 158·29-s + 60·31-s − 89·32-s + 16·34-s + 372·37-s + 170·38-s − 38·41-s + 516·43-s − 242·44-s + 124·46-s + 224·47-s − 606·49-s − 990·52-s + ⋯
L(s)  = 1  − 0.353·2-s − 1.37·4-s + 0.215·7-s + 0.662·8-s + 0.603·11-s + 1.92·13-s − 0.0763·14-s + 0.953·16-s − 0.228·17-s − 2.05·19-s − 0.213·22-s − 1.12·23-s − 0.678·26-s − 0.296·28-s + 1.01·29-s + 0.347·31-s − 0.491·32-s + 0.0807·34-s + 1.65·37-s + 0.725·38-s − 0.144·41-s + 1.82·43-s − 0.829·44-s + 0.397·46-s + 0.695·47-s − 1.76·49-s − 2.64·52-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6125625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6125625 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(6125625\)    =    \(3^{4} \cdot 5^{4} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(21324.6\)
Root analytic conductor: \(12.0842\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{2475} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 6125625,\ (\ :3/2, 3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(2.375092805\)
\(L(\frac12)\) \(\approx\) \(2.375092805\)
\(L(\frac{5}{2})\) 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$C_1$ \( ( 1 - p T )^{2} \)
good2$D_{4}$ \( 1 + T + 3 p^{2} T^{2} + p^{3} T^{3} + p^{6} T^{4} \)
7$D_{4}$ \( 1 - 4 T + 622 T^{2} - 4 p^{3} T^{3} + p^{6} T^{4} \)
13$D_{4}$ \( 1 - 90 T + 6402 T^{2} - 90 p^{3} T^{3} + p^{6} T^{4} \)
17$D_{4}$ \( 1 + 16 T + 1662 T^{2} + 16 p^{3} T^{3} + p^{6} T^{4} \)
19$D_{4}$ \( 1 + 170 T + 994 p T^{2} + 170 p^{3} T^{3} + p^{6} T^{4} \)
23$D_{4}$ \( 1 + 124 T + 12878 T^{2} + 124 p^{3} T^{3} + p^{6} T^{4} \)
29$D_{4}$ \( 1 - 158 T + 50106 T^{2} - 158 p^{3} T^{3} + p^{6} T^{4} \)
31$D_{4}$ \( 1 - 60 T + 59870 T^{2} - 60 p^{3} T^{3} + p^{6} T^{4} \)
37$D_{4}$ \( 1 - 372 T + 82590 T^{2} - 372 p^{3} T^{3} + p^{6} T^{4} \)
41$D_{4}$ \( 1 + 38 T + 37410 T^{2} + 38 p^{3} T^{3} + p^{6} T^{4} \)
43$D_{4}$ \( 1 - 12 p T + 160230 T^{2} - 12 p^{4} T^{3} + p^{6} T^{4} \)
47$D_{4}$ \( 1 - 224 T + 466 p T^{2} - 224 p^{3} T^{3} + p^{6} T^{4} \)
53$D_{4}$ \( 1 - 472 T + 190182 T^{2} - 472 p^{3} T^{3} + p^{6} T^{4} \)
59$D_{4}$ \( 1 + 248 T + 181334 T^{2} + 248 p^{3} T^{3} + p^{6} T^{4} \)
61$D_{4}$ \( 1 - 72 T - 107850 T^{2} - 72 p^{3} T^{3} + p^{6} T^{4} \)
67$D_{4}$ \( 1 - 744 T + 738822 T^{2} - 744 p^{3} T^{3} + p^{6} T^{4} \)
71$D_{4}$ \( 1 + 2060 T + 1768494 T^{2} + 2060 p^{3} T^{3} + p^{6} T^{4} \)
73$D_{4}$ \( 1 - 486 T + 822786 T^{2} - 486 p^{3} T^{3} + p^{6} T^{4} \)
79$D_{4}$ \( 1 - 642 T + 691166 T^{2} - 642 p^{3} T^{3} + p^{6} T^{4} \)
83$D_{4}$ \( 1 + 286 T + 392750 T^{2} + 286 p^{3} T^{3} + p^{6} T^{4} \)
89$D_{4}$ \( 1 + 244 T + 1355190 T^{2} + 244 p^{3} T^{3} + p^{6} T^{4} \)
97$D_{4}$ \( 1 - 168 T + 1053870 T^{2} - 168 p^{3} T^{3} + p^{6} T^{4} \)
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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

−8.655550900555565269075457068098, −8.534835576698951568378966764115, −8.045368204859701922825803387469, −8.036335358098949102577723822730, −7.32649292440202265912358086155, −6.76086911602870000457526118779, −6.28414438817192158100905695792, −6.17731322396329652214675722492, −5.78112780741648618080260284891, −5.22933009799415600614163347871, −4.61195791143290306025820003694, −4.25544151534436078484081468657, −4.01160124147327670174227973821, −3.91966804051084289133606233477, −2.99933043873008434671918242138, −2.59073616851752872733235022870, −1.79489723897345628325452278523, −1.45701157626238585353517137963, −0.61343271687873287247476512238, −0.56781221964193608466711584279, 0.56781221964193608466711584279, 0.61343271687873287247476512238, 1.45701157626238585353517137963, 1.79489723897345628325452278523, 2.59073616851752872733235022870, 2.99933043873008434671918242138, 3.91966804051084289133606233477, 4.01160124147327670174227973821, 4.25544151534436078484081468657, 4.61195791143290306025820003694, 5.22933009799415600614163347871, 5.78112780741648618080260284891, 6.17731322396329652214675722492, 6.28414438817192158100905695792, 6.76086911602870000457526118779, 7.32649292440202265912358086155, 8.036335358098949102577723822730, 8.045368204859701922825803387469, 8.534835576698951568378966764115, 8.655550900555565269075457068098

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