Properties

Label 4-336e2-1.1-c3e2-0-16
Degree $4$
Conductor $112896$
Sign $1$
Analytic cond. $393.016$
Root an. cond. $4.45248$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 6·3-s + 14·5-s − 14·7-s + 27·9-s − 18·11-s + 48·13-s − 84·15-s + 34·17-s + 16·19-s + 84·21-s − 110·23-s + 74·25-s − 108·27-s + 212·29-s + 136·31-s + 108·33-s − 196·35-s − 24·37-s − 288·39-s + 694·41-s + 584·43-s + 378·45-s + 316·47-s + 147·49-s − 204·51-s + 560·53-s − 252·55-s + ⋯
L(s)  = 1  − 1.15·3-s + 1.25·5-s − 0.755·7-s + 9-s − 0.493·11-s + 1.02·13-s − 1.44·15-s + 0.485·17-s + 0.193·19-s + 0.872·21-s − 0.997·23-s + 0.591·25-s − 0.769·27-s + 1.35·29-s + 0.787·31-s + 0.569·33-s − 0.946·35-s − 0.106·37-s − 1.18·39-s + 2.64·41-s + 2.07·43-s + 1.25·45-s + 0.980·47-s + 3/7·49-s − 0.560·51-s + 1.45·53-s − 0.617·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 112896 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112896 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(112896\)    =    \(2^{8} \cdot 3^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(393.016\)
Root analytic conductor: \(4.45248\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{336} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 112896,\ (\ :3/2, 3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(2.372302917\)
\(L(\frac12)\) \(\approx\) \(2.372302917\)
\(L(\frac{5}{2})\) 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$C_1$ \( ( 1 + p T )^{2} \)
good5$D_{4}$ \( 1 - 14 T + 122 T^{2} - 14 p^{3} T^{3} + p^{6} T^{4} \)
11$D_{4}$ \( 1 + 18 T + 1150 T^{2} + 18 p^{3} T^{3} + p^{6} T^{4} \)
13$D_{4}$ \( 1 - 48 T + 4262 T^{2} - 48 p^{3} T^{3} + p^{6} T^{4} \)
17$D_{4}$ \( 1 - 2 p T - 4222 T^{2} - 2 p^{4} T^{3} + p^{6} T^{4} \)
19$D_{4}$ \( 1 - 16 T + 10950 T^{2} - 16 p^{3} T^{3} + p^{6} T^{4} \)
23$D_{4}$ \( 1 + 110 T + 18686 T^{2} + 110 p^{3} T^{3} + p^{6} T^{4} \)
29$D_{4}$ \( 1 - 212 T + 57182 T^{2} - 212 p^{3} T^{3} + p^{6} T^{4} \)
31$D_{4}$ \( 1 - 136 T + 61374 T^{2} - 136 p^{3} T^{3} + p^{6} T^{4} \)
37$D_{4}$ \( 1 + 24 T - 18202 T^{2} + 24 p^{3} T^{3} + p^{6} T^{4} \)
41$D_{4}$ \( 1 - 694 T + 243914 T^{2} - 694 p^{3} T^{3} + p^{6} T^{4} \)
43$D_{4}$ \( 1 - 584 T + 232950 T^{2} - 584 p^{3} T^{3} + p^{6} T^{4} \)
47$D_{4}$ \( 1 - 316 T + 231902 T^{2} - 316 p^{3} T^{3} + p^{6} T^{4} \)
53$D_{4}$ \( 1 - 560 T + 369782 T^{2} - 560 p^{3} T^{3} + p^{6} T^{4} \)
59$D_{4}$ \( 1 - 492 T + 351622 T^{2} - 492 p^{3} T^{3} + p^{6} T^{4} \)
61$D_{4}$ \( 1 + 604 T + 406398 T^{2} + 604 p^{3} T^{3} + p^{6} T^{4} \)
67$D_{4}$ \( 1 - 1020 T + 843926 T^{2} - 1020 p^{3} T^{3} + p^{6} T^{4} \)
71$D_{4}$ \( 1 - 1710 T + 1336222 T^{2} - 1710 p^{3} T^{3} + p^{6} T^{4} \)
73$D_{4}$ \( 1 + 1312 T + 1201998 T^{2} + 1312 p^{3} T^{3} + p^{6} T^{4} \)
79$D_{4}$ \( 1 - 556 T + 751134 T^{2} - 556 p^{3} T^{3} + p^{6} T^{4} \)
83$D_{4}$ \( 1 - 264 T + 979750 T^{2} - 264 p^{3} T^{3} + p^{6} T^{4} \)
89$D_{4}$ \( 1 - 70 T - 186262 T^{2} - 70 p^{3} T^{3} + p^{6} T^{4} \)
97$D_{4}$ \( 1 + 136 T + 1812270 T^{2} + 136 p^{3} T^{3} + p^{6} T^{4} \)
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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

−11.05476892024899824471128444071, −10.98002244408899414124190480810, −10.33070418562649540252627902002, −10.17576136445753041002157695705, −9.450090557527880109447794445201, −9.394220373355558616586786959219, −8.607465162483739594912514717866, −7.983283756026743002968805068478, −7.50339840224651441394702483979, −6.78592024016844230763986035465, −6.25560152263658682681940921167, −6.10227937039976936883793034324, −5.46452972285355330213258282628, −5.27478869392472973071091551291, −4.08857261527020659674554083836, −4.02279590241151120267156786980, −2.71230447774277872997601667317, −2.36911295757533959499289115248, −1.14115428514291796534692255513, −0.70873087346276910385381314990, 0.70873087346276910385381314990, 1.14115428514291796534692255513, 2.36911295757533959499289115248, 2.71230447774277872997601667317, 4.02279590241151120267156786980, 4.08857261527020659674554083836, 5.27478869392472973071091551291, 5.46452972285355330213258282628, 6.10227937039976936883793034324, 6.25560152263658682681940921167, 6.78592024016844230763986035465, 7.50339840224651441394702483979, 7.983283756026743002968805068478, 8.607465162483739594912514717866, 9.394220373355558616586786959219, 9.450090557527880109447794445201, 10.17576136445753041002157695705, 10.33070418562649540252627902002, 10.98002244408899414124190480810, 11.05476892024899824471128444071

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