Properties

Label 4-1575e2-1.1-c3e2-0-7
Degree $4$
Conductor $2480625$
Sign $1$
Analytic cond. $8635.61$
Root an. cond. $9.63991$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·2-s + 5·4-s − 14·7-s − 27·8-s + 6·11-s − 16·13-s + 42·14-s + 69·16-s − 6·17-s + 64·19-s − 18·22-s + 6·23-s + 48·26-s − 70·28-s + 252·29-s + 40·31-s − 27·32-s + 18·34-s + 248·37-s − 192·38-s + 450·41-s − 376·43-s + 30·44-s − 18·46-s − 12·47-s + 147·49-s − 80·52-s + ⋯
L(s)  = 1  − 1.06·2-s + 5/8·4-s − 0.755·7-s − 1.19·8-s + 0.164·11-s − 0.341·13-s + 0.801·14-s + 1.07·16-s − 0.0856·17-s + 0.772·19-s − 0.174·22-s + 0.0543·23-s + 0.362·26-s − 0.472·28-s + 1.61·29-s + 0.231·31-s − 0.149·32-s + 0.0907·34-s + 1.10·37-s − 0.819·38-s + 1.71·41-s − 1.33·43-s + 0.102·44-s − 0.0576·46-s − 0.0372·47-s + 3/7·49-s − 0.213·52-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2480625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2480625 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(2480625\)    =    \(3^{4} \cdot 5^{4} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(8635.61\)
Root analytic conductor: \(9.63991\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1575} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 2480625,\ (\ :3/2, 3/2),\ 1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) 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 \( 1 \)
7$C_1$ \( ( 1 + p T )^{2} \)
good2$D_{4}$ \( 1 + 3 T + p^{2} T^{2} + 3 p^{3} T^{3} + p^{6} T^{4} \)
11$D_{4}$ \( 1 - 6 T + 1246 T^{2} - 6 p^{3} T^{3} + p^{6} T^{4} \)
13$D_{4}$ \( 1 + 16 T + 2406 T^{2} + 16 p^{3} T^{3} + p^{6} T^{4} \)
17$D_{4}$ \( 1 + 6 T + 9778 T^{2} + 6 p^{3} T^{3} + p^{6} T^{4} \)
19$D_{4}$ \( 1 - 64 T + 6534 T^{2} - 64 p^{3} T^{3} + p^{6} T^{4} \)
23$D_{4}$ \( 1 - 6 T + 7870 T^{2} - 6 p^{3} T^{3} + p^{6} T^{4} \)
29$D_{4}$ \( 1 - 252 T + 56446 T^{2} - 252 p^{3} T^{3} + p^{6} T^{4} \)
31$D_{4}$ \( 1 - 40 T - 13890 T^{2} - 40 p^{3} T^{3} + p^{6} T^{4} \)
37$D_{4}$ \( 1 - 248 T + 98214 T^{2} - 248 p^{3} T^{3} + p^{6} T^{4} \)
41$D_{4}$ \( 1 - 450 T + 175642 T^{2} - 450 p^{3} T^{3} + p^{6} T^{4} \)
43$D_{4}$ \( 1 + 376 T + 161526 T^{2} + 376 p^{3} T^{3} + p^{6} T^{4} \)
47$D_{4}$ \( 1 + 12 T + 141790 T^{2} + 12 p^{3} T^{3} + p^{6} T^{4} \)
53$D_{4}$ \( 1 + 1104 T + 602230 T^{2} + 1104 p^{3} T^{3} + p^{6} T^{4} \)
59$D_{4}$ \( 1 + 804 T + 380614 T^{2} + 804 p^{3} T^{3} + p^{6} T^{4} \)
61$D_{4}$ \( 1 + 428 T + 425886 T^{2} + 428 p^{3} T^{3} + p^{6} T^{4} \)
67$D_{4}$ \( 1 + 148 T + 440790 T^{2} + 148 p^{3} T^{3} + p^{6} T^{4} \)
71$D_{4}$ \( 1 + 954 T + 13106 p T^{2} + 954 p^{3} T^{3} + p^{6} T^{4} \)
73$D_{4}$ \( 1 + 1072 T + 1063278 T^{2} + 1072 p^{3} T^{3} + p^{6} T^{4} \)
79$D_{4}$ \( 1 + 572 T + 901662 T^{2} + 572 p^{3} T^{3} + p^{6} T^{4} \)
83$D_{4}$ \( 1 - 1944 T + 1957030 T^{2} - 1944 p^{3} T^{3} + p^{6} T^{4} \)
89$D_{4}$ \( 1 + 366 T + 1156090 T^{2} + 366 p^{3} T^{3} + p^{6} T^{4} \)
97$D_{4}$ \( 1 + 808 T + 903054 T^{2} + 808 p^{3} T^{3} + p^{6} T^{4} \)
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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

−8.932068889520827437621225974750, −8.622761295756656614129980179154, −8.048444898566236292797922599698, −7.73345263277985193941617836602, −7.43320567989859671916110048646, −6.88794073984370008804533430532, −6.32989067570290056969484121359, −6.13139433052273228152951024027, −5.97855626316292299961254700062, −5.11529370742195282845364745914, −4.62644258946915810473079426420, −4.37389205761368587742624946871, −3.46735538989211370017006397673, −2.97182062602480805480866852107, −2.92833208023856057722631891038, −2.20221708205696867167037112828, −1.32205003009627556106295966332, −1.00068219456382963485289928963, 0, 0, 1.00068219456382963485289928963, 1.32205003009627556106295966332, 2.20221708205696867167037112828, 2.92833208023856057722631891038, 2.97182062602480805480866852107, 3.46735538989211370017006397673, 4.37389205761368587742624946871, 4.62644258946915810473079426420, 5.11529370742195282845364745914, 5.97855626316292299961254700062, 6.13139433052273228152951024027, 6.32989067570290056969484121359, 6.88794073984370008804533430532, 7.43320567989859671916110048646, 7.73345263277985193941617836602, 8.048444898566236292797922599698, 8.622761295756656614129980179154, 8.932068889520827437621225974750

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