Properties

Label 4-1904e2-1.1-c1e2-0-3
Degree $4$
Conductor $3625216$
Sign $1$
Analytic cond. $231.146$
Root an. cond. $3.89916$
Motivic weight $1$
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  + 3-s − 5-s + 2·7-s − 4·9-s + 6·11-s − 2·13-s − 15-s + 2·17-s + 8·19-s + 2·21-s + 2·23-s − 8·25-s − 6·27-s + 2·29-s + 3·31-s + 6·33-s − 2·35-s − 8·37-s − 2·39-s + 11·41-s + 13·43-s + 4·45-s + 8·47-s + 3·49-s + 2·51-s − 3·53-s − 6·55-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.447·5-s + 0.755·7-s − 4/3·9-s + 1.80·11-s − 0.554·13-s − 0.258·15-s + 0.485·17-s + 1.83·19-s + 0.436·21-s + 0.417·23-s − 8/5·25-s − 1.15·27-s + 0.371·29-s + 0.538·31-s + 1.04·33-s − 0.338·35-s − 1.31·37-s − 0.320·39-s + 1.71·41-s + 1.98·43-s + 0.596·45-s + 1.16·47-s + 3/7·49-s + 0.280·51-s − 0.412·53-s − 0.809·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3625216 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3625216 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(3625216\)    =    \(2^{8} \cdot 7^{2} \cdot 17^{2}\)
Sign: $1$
Analytic conductor: \(231.146\)
Root analytic conductor: \(3.89916\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 3625216,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.411962237\)
\(L(\frac12)\) \(\approx\) \(3.411962237\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7$C_1$ \( ( 1 - T )^{2} \)
17$C_1$ \( ( 1 - T )^{2} \)
good3$D_{4}$ \( 1 - T + 5 T^{2} - p T^{3} + p^{2} T^{4} \)
5$D_{4}$ \( 1 + T + 9 T^{2} + p T^{3} + p^{2} T^{4} \)
11$C_4$ \( 1 - 6 T + 26 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
13$D_{4}$ \( 1 + 2 T + 22 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
19$C_4$ \( 1 - 8 T + 34 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
23$C_2^2$ \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 - 2 T + 14 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 - 3 T + 3 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 + 8 T + 70 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 - 11 T + 81 T^{2} - 11 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 - 13 T + 97 T^{2} - 13 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 - 8 T + 90 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 + 3 T - 43 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
59$C_2$ \( ( 1 + p T^{2} )^{2} \)
61$D_{4}$ \( 1 - T + p T^{2} - p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 - 15 T + 159 T^{2} - 15 p T^{3} + p^{2} T^{4} \)
71$C_4$ \( 1 + 4 T + 66 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 - 21 T + 245 T^{2} - 21 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 - 12 T + 114 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 + 4 T - 10 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
97$D_{4}$ \( 1 - 19 T + 253 T^{2} - 19 p T^{3} + p^{2} T^{4} \)
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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

−9.251494114526931680146744358608, −9.209935207095168281070748925085, −8.557481013191392824325509638094, −8.258022563354538671012713496604, −7.85145214379333741567846646103, −7.52270453110405666428798929251, −7.27223624438057612688824540281, −6.66941540525329587574385984930, −6.19537584911371863983466434923, −5.76677596764495562855844723021, −5.37840242461091345403321708712, −5.06491180027596975030854899788, −4.31149189186211904874032482224, −4.02144685807586174488539976853, −3.44193973581641863921648003515, −3.26944270530189560424564744998, −2.35049339893568582670634373650, −2.23667949207185852046902020726, −1.17578664884106330615335621997, −0.75770666601970373686798374340, 0.75770666601970373686798374340, 1.17578664884106330615335621997, 2.23667949207185852046902020726, 2.35049339893568582670634373650, 3.26944270530189560424564744998, 3.44193973581641863921648003515, 4.02144685807586174488539976853, 4.31149189186211904874032482224, 5.06491180027596975030854899788, 5.37840242461091345403321708712, 5.76677596764495562855844723021, 6.19537584911371863983466434923, 6.66941540525329587574385984930, 7.27223624438057612688824540281, 7.52270453110405666428798929251, 7.85145214379333741567846646103, 8.258022563354538671012713496604, 8.557481013191392824325509638094, 9.209935207095168281070748925085, 9.251494114526931680146744358608

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