Properties

Label 4-1216e2-1.1-c1e2-0-26
Degree $4$
Conductor $1478656$
Sign $1$
Analytic cond. $94.2803$
Root an. cond. $3.11605$
Motivic weight $1$
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  + 3-s + 5-s − 6·7-s − 9-s − 3·11-s − 13-s + 15-s − 8·17-s + 2·19-s − 6·21-s − 7·23-s − 5·25-s + 3·29-s − 10·31-s − 3·33-s − 6·35-s − 6·37-s − 39-s + 8·41-s + 15·43-s − 45-s − 5·47-s + 13·49-s − 8·51-s + 13·53-s − 3·55-s + 2·57-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.447·5-s − 2.26·7-s − 1/3·9-s − 0.904·11-s − 0.277·13-s + 0.258·15-s − 1.94·17-s + 0.458·19-s − 1.30·21-s − 1.45·23-s − 25-s + 0.557·29-s − 1.79·31-s − 0.522·33-s − 1.01·35-s − 0.986·37-s − 0.160·39-s + 1.24·41-s + 2.28·43-s − 0.149·45-s − 0.729·47-s + 13/7·49-s − 1.12·51-s + 1.78·53-s − 0.404·55-s + 0.264·57-s + ⋯

Functional equation

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

Invariants

Degree: \(4\)
Conductor: \(1478656\)    =    \(2^{12} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(94.2803\)
Root analytic conductor: \(3.11605\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1216} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 1478656,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 - T )^{2} \)
good3$D_{4}$ \( 1 - T + 2 T^{2} - p T^{3} + p^{2} T^{4} \)
5$D_{4}$ \( 1 - T + 6 T^{2} - p T^{3} + p^{2} T^{4} \)
7$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
11$D_{4}$ \( 1 + 3 T + 20 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
13$D_{4}$ \( 1 + T + 22 T^{2} + p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 + 8 T + 33 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
23$D_{4}$ \( 1 + 7 T + 54 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 - 3 T + 22 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 + 10 T + 70 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 + 6 T + 66 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
41$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
43$D_{4}$ \( 1 - 15 T + 138 T^{2} - 15 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 + 5 T + 62 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 - 13 T + 144 T^{2} - 13 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 + 13 T + 156 T^{2} + 13 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 + 9 T + 36 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 + 5 T + 136 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 + 8 T + 90 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 + 2 T + 79 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
79$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
83$D_{4}$ \( 1 + 10 T + 38 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 + 6 T + 34 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 - 6 T + 186 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
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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

−9.401534569465140448478223180627, −9.226161577616968284009900057929, −8.820604068959892843241852593888, −8.526408412427992663998726214108, −7.76902922240119736777179286340, −7.45554932226532611543732923011, −7.13039175006366889637193122444, −6.55352104764641269045717217135, −6.15930323606434666381551406606, −5.74448315765810847195685347969, −5.62603650770663543413790010552, −4.70974977591438141331643939261, −4.10152857220081832419368412288, −3.89522227767074913916875693294, −2.98956207569121307769104942490, −2.87367757091120813503216685532, −2.34497001915234297508936010427, −1.70831221281079586954224147811, 0, 0, 1.70831221281079586954224147811, 2.34497001915234297508936010427, 2.87367757091120813503216685532, 2.98956207569121307769104942490, 3.89522227767074913916875693294, 4.10152857220081832419368412288, 4.70974977591438141331643939261, 5.62603650770663543413790010552, 5.74448315765810847195685347969, 6.15930323606434666381551406606, 6.55352104764641269045717217135, 7.13039175006366889637193122444, 7.45554932226532611543732923011, 7.76902922240119736777179286340, 8.526408412427992663998726214108, 8.820604068959892843241852593888, 9.226161577616968284009900057929, 9.401534569465140448478223180627

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