Properties

Label 4-112e2-1.1-c3e2-0-4
Degree $4$
Conductor $12544$
Sign $1$
Analytic cond. $43.6684$
Root an. cond. $2.57064$
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  − 3-s − 9·5-s + 28·7-s + 27·9-s + 45·11-s + 9·15-s + 63·17-s + 17·19-s − 28·21-s + 243·23-s − 71·25-s − 80·27-s + 180·29-s − 17·31-s − 45·33-s − 252·35-s − 199·37-s − 243·45-s − 567·47-s + 441·49-s − 63·51-s + 333·53-s − 405·55-s − 17·57-s − 801·59-s − 621·61-s + 756·63-s + ⋯
L(s)  = 1  − 0.192·3-s − 0.804·5-s + 1.51·7-s + 9-s + 1.23·11-s + 0.154·15-s + 0.898·17-s + 0.205·19-s − 0.290·21-s + 2.20·23-s − 0.567·25-s − 0.570·27-s + 1.15·29-s − 0.0984·31-s − 0.237·33-s − 1.21·35-s − 0.884·37-s − 0.804·45-s − 1.75·47-s + 9/7·49-s − 0.172·51-s + 0.863·53-s − 0.992·55-s − 0.0395·57-s − 1.76·59-s − 1.30·61-s + 1.51·63-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 12544 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12544 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(12544\)    =    \(2^{8} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(43.6684\)
Root analytic conductor: \(2.57064\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{112} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 12544,\ (\ :3/2, 3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(2.542116557\)
\(L(\frac12)\) \(\approx\) \(2.542116557\)
\(L(\frac{5}{2})\) 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 \( 1 \)
7$C_2$ \( 1 - 4 p T + p^{3} T^{2} \)
good3$C_2^2$ \( 1 + T - 26 T^{2} + p^{3} T^{3} + p^{6} T^{4} \)
5$C_2^2$ \( 1 + 9 T + 152 T^{2} + 9 p^{3} T^{3} + p^{6} T^{4} \)
11$C_2^2$ \( 1 - 45 T + 2006 T^{2} - 45 p^{3} T^{3} + p^{6} T^{4} \)
13$C_2^2$ \( 1 + 406 T^{2} + p^{6} T^{4} \)
17$C_2^2$ \( 1 - 63 T + 6236 T^{2} - 63 p^{3} T^{3} + p^{6} T^{4} \)
19$C_2^2$ \( 1 - 17 T - 6570 T^{2} - 17 p^{3} T^{3} + p^{6} T^{4} \)
23$C_2^2$ \( 1 - 243 T + 31850 T^{2} - 243 p^{3} T^{3} + p^{6} T^{4} \)
29$C_2$ \( ( 1 - 90 T + p^{3} T^{2} )^{2} \)
31$C_2^2$ \( 1 + 17 T - 29502 T^{2} + 17 p^{3} T^{3} + p^{6} T^{4} \)
37$C_2^2$ \( 1 + 199 T - 11052 T^{2} + 199 p^{3} T^{3} + p^{6} T^{4} \)
41$C_2^2$ \( 1 - 102850 T^{2} + p^{6} T^{4} \)
43$C_2^2$ \( 1 - 95066 T^{2} + p^{6} T^{4} \)
47$C_2^2$ \( 1 + 567 T + 217666 T^{2} + 567 p^{3} T^{3} + p^{6} T^{4} \)
53$C_2^2$ \( 1 - 333 T - 37988 T^{2} - 333 p^{3} T^{3} + p^{6} T^{4} \)
59$C_2^2$ \( 1 + 801 T + 436222 T^{2} + 801 p^{3} T^{3} + p^{6} T^{4} \)
61$C_2^2$ \( 1 + 621 T + 355528 T^{2} + 621 p^{3} T^{3} + p^{6} T^{4} \)
67$C_2^2$ \( 1 + 375 T + 347638 T^{2} + 375 p^{3} T^{3} + p^{6} T^{4} \)
71$C_2^2$ \( 1 - 477250 T^{2} + p^{6} T^{4} \)
73$C_2^2$ \( 1 - 699 T + 551884 T^{2} - 699 p^{3} T^{3} + p^{6} T^{4} \)
79$C_2^2$ \( 1 - 2067 T + 1917202 T^{2} - 2067 p^{3} T^{3} + p^{6} T^{4} \)
83$C_2$ \( ( 1 + 468 T + p^{3} T^{2} )^{2} \)
89$C_2^2$ \( 1 + 333 T + 741932 T^{2} + 333 p^{3} T^{3} + p^{6} T^{4} \)
97$C_2^2$ \( 1 + 113902 T^{2} + p^{6} T^{4} \)
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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

−13.43889098003630781318289833507, −12.76179935428352140991660546308, −12.05115306550780090487314236709, −12.03794186359490671223511060122, −11.17753660438961887839456111288, −11.11812697405302105441483570519, −10.28903657111439428336160040551, −9.733009354133742288989243740934, −8.984234520350558678377662665603, −8.593257525884054257091566411157, −7.65217838418125466251495668060, −7.60619864054561974000385032259, −6.81177956970681326298237333720, −6.15203693084191986172668840553, −4.92355969118957217379179870001, −4.88325203345011471344931012578, −3.94522498795801371093276684657, −3.24835159982206457382141625806, −1.67673324021216051478958280464, −1.03297301311748921299127165765, 1.03297301311748921299127165765, 1.67673324021216051478958280464, 3.24835159982206457382141625806, 3.94522498795801371093276684657, 4.88325203345011471344931012578, 4.92355969118957217379179870001, 6.15203693084191986172668840553, 6.81177956970681326298237333720, 7.60619864054561974000385032259, 7.65217838418125466251495668060, 8.593257525884054257091566411157, 8.984234520350558678377662665603, 9.733009354133742288989243740934, 10.28903657111439428336160040551, 11.11812697405302105441483570519, 11.17753660438961887839456111288, 12.03794186359490671223511060122, 12.05115306550780090487314236709, 12.76179935428352140991660546308, 13.43889098003630781318289833507

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