Properties

Label 4-112e2-1.1-c1e2-0-2
Degree $4$
Conductor $12544$
Sign $1$
Analytic cond. $0.799816$
Root an. cond. $0.945687$
Motivic weight $1$
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  − 2·2-s + 2·4-s + 4·5-s − 8·10-s + 2·11-s − 4·16-s − 4·17-s + 4·19-s + 8·20-s − 4·22-s + 8·25-s + 14·29-s − 16·31-s + 8·32-s + 8·34-s − 10·37-s − 8·38-s − 2·43-s + 4·44-s − 24·47-s − 49-s − 16·50-s − 2·53-s + 8·55-s − 28·58-s + 16·59-s + 12·61-s + ⋯
L(s)  = 1  − 1.41·2-s + 4-s + 1.78·5-s − 2.52·10-s + 0.603·11-s − 16-s − 0.970·17-s + 0.917·19-s + 1.78·20-s − 0.852·22-s + 8/5·25-s + 2.59·29-s − 2.87·31-s + 1.41·32-s + 1.37·34-s − 1.64·37-s − 1.29·38-s − 0.304·43-s + 0.603·44-s − 3.50·47-s − 1/7·49-s − 2.26·50-s − 0.274·53-s + 1.07·55-s − 3.67·58-s + 2.08·59-s + 1.53·61-s + ⋯

Functional equation

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

Invariants

Degree: \(4\)
Conductor: \(12544\)    =    \(2^{8} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(0.799816\)
Root analytic conductor: \(0.945687\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{112} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 12544,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7114488872\)
\(L(\frac12)\) \(\approx\) \(0.7114488872\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + p T + p T^{2} \)
7$C_2$ \( 1 + T^{2} \)
good3$C_2^2$ \( 1 + p^{2} T^{4} \)
5$C_2^2$ \( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
11$C_2^2$ \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
13$C_2^2$ \( 1 + p^{2} T^{4} \)
17$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
19$C_2^2$ \( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
23$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 - 4 T + p T^{2} ) \)
31$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
41$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
43$C_2^2$ \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
47$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
53$C_2^2$ \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 - 16 T + 128 T^{2} - 16 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 - 12 T + 72 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 - p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
79$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 + 20 T + 200 T^{2} + 20 p T^{3} + p^{2} T^{4} \)
89$C_2^2$ \( 1 + 18 T^{2} + p^{2} T^{4} \)
97$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
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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

−14.07709379257313501434929525229, −13.21080224703493065311584299987, −13.10435677069817317715597291629, −12.33374275476837685879156416076, −11.41893359581473435188152530904, −11.26811707037460048297898716732, −10.42474346384492172731049067434, −10.00937462620166698547838782373, −9.683924366003053898879873910454, −9.197203073722881335501337628871, −8.540009028955318294445626349058, −8.332459713466063053230675288684, −7.11194844754978421779119417990, −6.84081957360369013901511394052, −6.27061469703395830923553716435, −5.33221791626940470964482586284, −4.84926899829197202780201987783, −3.49326979522568431710341003998, −2.23309122526353656923885425256, −1.51802328772081144901566458296, 1.51802328772081144901566458296, 2.23309122526353656923885425256, 3.49326979522568431710341003998, 4.84926899829197202780201987783, 5.33221791626940470964482586284, 6.27061469703395830923553716435, 6.84081957360369013901511394052, 7.11194844754978421779119417990, 8.332459713466063053230675288684, 8.540009028955318294445626349058, 9.197203073722881335501337628871, 9.683924366003053898879873910454, 10.00937462620166698547838782373, 10.42474346384492172731049067434, 11.26811707037460048297898716732, 11.41893359581473435188152530904, 12.33374275476837685879156416076, 13.10435677069817317715597291629, 13.21080224703493065311584299987, 14.07709379257313501434929525229

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