Properties

Label 4-1216e2-1.1-c3e2-0-2
Degree $4$
Conductor $1478656$
Sign $1$
Analytic cond. $5147.53$
Root an. cond. $8.47032$
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  + 9·3-s + 9·5-s + 18·7-s + 25·9-s − 17·11-s − 17·13-s + 81·15-s − 80·17-s + 38·19-s + 162·21-s − 73·23-s − 25·25-s − 36·27-s − 3·29-s − 212·31-s − 153·33-s + 162·35-s − 192·37-s − 153·39-s − 50·41-s + 677·43-s + 225·45-s + 389·47-s − 151·49-s − 720·51-s + 1.21e3·53-s − 153·55-s + ⋯
L(s)  = 1  + 1.73·3-s + 0.804·5-s + 0.971·7-s + 0.925·9-s − 0.465·11-s − 0.362·13-s + 1.39·15-s − 1.14·17-s + 0.458·19-s + 1.68·21-s − 0.661·23-s − 1/5·25-s − 0.256·27-s − 0.0192·29-s − 1.22·31-s − 0.807·33-s + 0.782·35-s − 0.853·37-s − 0.628·39-s − 0.190·41-s + 2.40·43-s + 0.745·45-s + 1.20·47-s − 0.440·49-s − 1.97·51-s + 3.15·53-s − 0.375·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1478656 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1478656 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1478656\)    =    \(2^{12} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(5147.53\)
Root analytic conductor: \(8.47032\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1216} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1478656,\ (\ :3/2, 3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(6.153243054\)
\(L(\frac12)\) \(\approx\) \(6.153243054\)
\(L(\frac{5}{2})\) 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)$
bad2 \( 1 \)
19$C_1$ \( ( 1 - p T )^{2} \)
good3$D_{4}$ \( 1 - p^{2} T + 56 T^{2} - p^{5} T^{3} + p^{6} T^{4} \)
5$D_{4}$ \( 1 - 9 T + 106 T^{2} - 9 p^{3} T^{3} + p^{6} T^{4} \)
7$D_{4}$ \( 1 - 18 T + 475 T^{2} - 18 p^{3} T^{3} + p^{6} T^{4} \)
11$D_{4}$ \( 1 + 17 T + 2716 T^{2} + 17 p^{3} T^{3} + p^{6} T^{4} \)
13$D_{4}$ \( 1 + 17 T + 1382 T^{2} + 17 p^{3} T^{3} + p^{6} T^{4} \)
17$D_{4}$ \( 1 + 80 T + 11353 T^{2} + 80 p^{3} T^{3} + p^{6} T^{4} \)
23$D_{4}$ \( 1 + 73 T + 22582 T^{2} + 73 p^{3} T^{3} + p^{6} T^{4} \)
29$D_{4}$ \( 1 + 3 T + 40732 T^{2} + 3 p^{3} T^{3} + p^{6} T^{4} \)
31$D_{4}$ \( 1 + 212 T + 35486 T^{2} + 212 p^{3} T^{3} + p^{6} T^{4} \)
37$D_{4}$ \( 1 + 192 T + 96214 T^{2} + 192 p^{3} T^{3} + p^{6} T^{4} \)
41$D_{4}$ \( 1 + 50 T + 136642 T^{2} + 50 p^{3} T^{3} + p^{6} T^{4} \)
43$D_{4}$ \( 1 - 677 T + 272702 T^{2} - 677 p^{3} T^{3} + p^{6} T^{4} \)
47$D_{4}$ \( 1 - 389 T + 153478 T^{2} - 389 p^{3} T^{3} + p^{6} T^{4} \)
53$D_{4}$ \( 1 - 23 p T + 663970 T^{2} - 23 p^{4} T^{3} + p^{6} T^{4} \)
59$D_{4}$ \( 1 + 287 T + 419944 T^{2} + 287 p^{3} T^{3} + p^{6} T^{4} \)
61$D_{4}$ \( 1 + 313 T + 253596 T^{2} + 313 p^{3} T^{3} + p^{6} T^{4} \)
67$D_{4}$ \( 1 - 1223 T + 867254 T^{2} - 1223 p^{3} T^{3} + p^{6} T^{4} \)
71$D_{4}$ \( 1 + 200 T + 480250 T^{2} + 200 p^{3} T^{3} + p^{6} T^{4} \)
73$D_{4}$ \( 1 - 378 T + 195883 T^{2} - 378 p^{3} T^{3} + p^{6} T^{4} \)
79$D_{4}$ \( 1 + 1350 T + 1380310 T^{2} + 1350 p^{3} T^{3} + p^{6} T^{4} \)
83$D_{4}$ \( 1 + 670 T + 568942 T^{2} + 670 p^{3} T^{3} + p^{6} T^{4} \)
89$D_{4}$ \( 1 + 236 T + 778834 T^{2} + 236 p^{3} T^{3} + p^{6} T^{4} \)
97$D_{4}$ \( 1 - 1294 T + 2054082 T^{2} - 1294 p^{3} T^{3} + p^{6} T^{4} \)
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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

−9.185973526962627826523905023070, −9.150603511959296760322710962791, −8.718663720873393263019731339468, −8.593488406694392387279716918853, −7.78582264149352925714818608840, −7.78038778777941103874677804663, −7.22620626873839103528204963063, −6.94130391336767487065544854205, −5.96159475927526350770924275753, −5.94547890690113992705228182606, −5.23147452779487324810306963110, −4.94639892186479683212488840222, −4.19435562168055639873312091549, −3.88634965986273758773453391693, −3.33954277348374511772005961027, −2.57393821642082024247090844101, −2.30606826442861438478680056216, −2.09126789578427975043930095052, −1.36612815329328445351625252446, −0.48303801987815796509169113044, 0.48303801987815796509169113044, 1.36612815329328445351625252446, 2.09126789578427975043930095052, 2.30606826442861438478680056216, 2.57393821642082024247090844101, 3.33954277348374511772005961027, 3.88634965986273758773453391693, 4.19435562168055639873312091549, 4.94639892186479683212488840222, 5.23147452779487324810306963110, 5.94547890690113992705228182606, 5.96159475927526350770924275753, 6.94130391336767487065544854205, 7.22620626873839103528204963063, 7.78038778777941103874677804663, 7.78582264149352925714818608840, 8.593488406694392387279716918853, 8.718663720873393263019731339468, 9.150603511959296760322710962791, 9.185973526962627826523905023070

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