Properties

Label 4-825e2-1.1-c3e2-0-3
Degree $4$
Conductor $680625$
Sign $1$
Analytic cond. $2369.40$
Root an. cond. $6.97686$
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 − 6·3-s − 11·4-s − 6·6-s + 4·7-s − 15·8-s + 27·9-s − 22·11-s + 66·12-s + 90·13-s + 4·14-s + 61·16-s + 16·17-s + 27·18-s − 170·19-s − 24·21-s − 22·22-s + 124·23-s + 90·24-s + 90·26-s − 108·27-s − 44·28-s − 158·29-s + 60·31-s + 89·32-s + 132·33-s + 16·34-s + ⋯
L(s)  = 1  + 0.353·2-s − 1.15·3-s − 1.37·4-s − 0.408·6-s + 0.215·7-s − 0.662·8-s + 9-s − 0.603·11-s + 1.58·12-s + 1.92·13-s + 0.0763·14-s + 0.953·16-s + 0.228·17-s + 0.353·18-s − 2.05·19-s − 0.249·21-s − 0.213·22-s + 1.12·23-s + 0.765·24-s + 0.678·26-s − 0.769·27-s − 0.296·28-s − 1.01·29-s + 0.347·31-s + 0.491·32-s + 0.696·33-s + 0.0807·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 680625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 680625 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(680625\)    =    \(3^{2} \cdot 5^{4} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(2369.40\)
Root analytic conductor: \(6.97686\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{825} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 680625,\ (\ :3/2, 3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(1.743813736\)
\(L(\frac12)\) \(\approx\) \(1.743813736\)
\(L(\frac{5}{2})\) 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$C_1$ \( ( 1 + p T )^{2} \)
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

−9.876528510536384947953267090958, −9.806986566435316226187738075720, −9.068030042095891331729383022713, −8.954080081240248141494269535549, −8.277786429051531621205711879598, −8.016825592674984079214629781883, −7.65570917597865097032426039793, −6.71788831850485457084068465706, −6.32318897000562416886075608313, −6.23415309639988120884263187498, −5.41503299734325727280676594738, −5.28967833031309285134040649406, −4.51972008784312345191272966679, −4.47846261270584869041862970750, −3.66192660395499297026734709011, −3.53585299880254668214246467297, −2.42307071631622425828895461978, −1.71164066558505707530058870290, −0.66978612872596675055054712958, −0.63797008900176234647170914173, 0.63797008900176234647170914173, 0.66978612872596675055054712958, 1.71164066558505707530058870290, 2.42307071631622425828895461978, 3.53585299880254668214246467297, 3.66192660395499297026734709011, 4.47846261270584869041862970750, 4.51972008784312345191272966679, 5.28967833031309285134040649406, 5.41503299734325727280676594738, 6.23415309639988120884263187498, 6.32318897000562416886075608313, 6.71788831850485457084068465706, 7.65570917597865097032426039793, 8.016825592674984079214629781883, 8.277786429051531621205711879598, 8.954080081240248141494269535549, 9.068030042095891331729383022713, 9.806986566435316226187738075720, 9.876528510536384947953267090958

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