Properties

Label 4-2352e2-1.1-c3e2-0-19
Degree $4$
Conductor $5531904$
Sign $1$
Analytic cond. $19257.8$
Root an. cond. $11.7801$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 6·3-s − 11·5-s + 27·9-s + 5·11-s − 5·13-s − 66·15-s − 100·17-s + 67·19-s − 76·23-s − 111·25-s + 108·27-s + 275·29-s + 362·31-s + 30·33-s − 5·37-s − 30·39-s + 162·41-s − 721·43-s − 297·45-s − 216·47-s − 600·51-s + 495·53-s − 55·55-s + 402·57-s + 173·59-s + 532·61-s + 55·65-s + ⋯
L(s)  = 1  + 1.15·3-s − 0.983·5-s + 9-s + 0.137·11-s − 0.106·13-s − 1.13·15-s − 1.42·17-s + 0.808·19-s − 0.689·23-s − 0.887·25-s + 0.769·27-s + 1.76·29-s + 2.09·31-s + 0.158·33-s − 0.0222·37-s − 0.123·39-s + 0.617·41-s − 2.55·43-s − 0.983·45-s − 0.670·47-s − 1.64·51-s + 1.28·53-s − 0.134·55-s + 0.934·57-s + 0.381·59-s + 1.11·61-s + 0.104·65-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5531904 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5531904 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(5531904\)    =    \(2^{8} \cdot 3^{2} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(19257.8\)
Root analytic conductor: \(11.7801\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{2352} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 5531904,\ (\ :3/2, 3/2),\ 1)\)

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)$
bad2 \( 1 \)
3$C_1$ \( ( 1 - p T )^{2} \)
7 \( 1 \)
good5$D_{4}$ \( 1 + 11 T + 232 T^{2} + 11 p^{3} T^{3} + p^{6} T^{4} \)
11$D_{4}$ \( 1 - 5 T + 304 T^{2} - 5 p^{3} T^{3} + p^{6} T^{4} \)
13$D_{4}$ \( 1 + 5 T + 3194 T^{2} + 5 p^{3} T^{3} + p^{6} T^{4} \)
17$D_{4}$ \( 1 + 100 T + 11554 T^{2} + 100 p^{3} T^{3} + p^{6} T^{4} \)
19$D_{4}$ \( 1 - 67 T + 14406 T^{2} - 67 p^{3} T^{3} + p^{6} T^{4} \)
23$D_{4}$ \( 1 + 76 T + 6478 T^{2} + 76 p^{3} T^{3} + p^{6} T^{4} \)
29$D_{4}$ \( 1 - 275 T + 61846 T^{2} - 275 p^{3} T^{3} + p^{6} T^{4} \)
31$D_{4}$ \( 1 - 362 T + 73043 T^{2} - 362 p^{3} T^{3} + p^{6} T^{4} \)
37$D_{4}$ \( 1 + 5 T + 3606 T^{2} + 5 p^{3} T^{3} + p^{6} T^{4} \)
41$D_{4}$ \( 1 - 162 T + 128770 T^{2} - 162 p^{3} T^{3} + p^{6} T^{4} \)
43$D_{4}$ \( 1 + 721 T + 288540 T^{2} + 721 p^{3} T^{3} + p^{6} T^{4} \)
47$D_{4}$ \( 1 + 216 T + 156778 T^{2} + 216 p^{3} T^{3} + p^{6} T^{4} \)
53$D_{4}$ \( 1 - 495 T + 355102 T^{2} - 495 p^{3} T^{3} + p^{6} T^{4} \)
59$D_{4}$ \( 1 - 173 T + 282706 T^{2} - 173 p^{3} T^{3} + p^{6} T^{4} \)
61$D_{4}$ \( 1 - 532 T + 447518 T^{2} - 532 p^{3} T^{3} + p^{6} T^{4} \)
67$D_{4}$ \( 1 + 111 T + 318532 T^{2} + 111 p^{3} T^{3} + p^{6} T^{4} \)
71$D_{4}$ \( 1 + 1600 T + 1262410 T^{2} + 1600 p^{3} T^{3} + p^{6} T^{4} \)
73$D_{4}$ \( 1 + 1215 T + 1011556 T^{2} + 1215 p^{3} T^{3} + p^{6} T^{4} \)
79$D_{4}$ \( 1 + 1460 T + 1333505 T^{2} + 1460 p^{3} T^{3} + p^{6} T^{4} \)
83$D_{4}$ \( 1 + 1409 T + 1147696 T^{2} + 1409 p^{3} T^{3} + p^{6} T^{4} \)
89$D_{4}$ \( 1 - 1974 T + 2298994 T^{2} - 1974 p^{3} T^{3} + p^{6} T^{4} \)
97$D_{4}$ \( 1 + 561 T + 1863448 T^{2} + 561 p^{3} T^{3} + p^{6} T^{4} \)
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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

−8.371265433724889973001824253773, −8.304531954622511075688480929681, −7.58875644631626325749891629200, −7.51812530491013254373458673871, −6.90228987848464914718375896584, −6.75414196103584944190352366841, −6.04849359702424076022020697274, −5.94900799166495831162353147796, −4.96856226153666463224181356574, −4.77055690943627350937488380148, −4.24420278218345820367824723616, −4.09313309230251420716759533840, −3.45963951248356663503328635068, −3.10395333161220451828094653506, −2.46679314217479905467348209222, −2.39107797986518227122124598247, −1.34012716247475349775739027010, −1.22597306770555361957734617123, 0, 0, 1.22597306770555361957734617123, 1.34012716247475349775739027010, 2.39107797986518227122124598247, 2.46679314217479905467348209222, 3.10395333161220451828094653506, 3.45963951248356663503328635068, 4.09313309230251420716759533840, 4.24420278218345820367824723616, 4.77055690943627350937488380148, 4.96856226153666463224181356574, 5.94900799166495831162353147796, 6.04849359702424076022020697274, 6.75414196103584944190352366841, 6.90228987848464914718375896584, 7.51812530491013254373458673871, 7.58875644631626325749891629200, 8.304531954622511075688480929681, 8.371265433724889973001824253773

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