Properties

Label 4-112e2-1.1-c3e2-0-7
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  + 7·3-s + 27·5-s − 28·7-s + 27·9-s + 21·11-s + 189·15-s − 57·17-s − 119·19-s − 196·21-s + 231·23-s + 361·25-s + 224·27-s + 420·29-s − 301·31-s + 147·33-s − 756·35-s + 77·37-s + 729·45-s + 357·47-s + 441·49-s − 399·51-s − 327·53-s + 567·55-s − 833·57-s − 609·59-s + 1.19e3·61-s − 756·63-s + ⋯
L(s)  = 1  + 1.34·3-s + 2.41·5-s − 1.51·7-s + 9-s + 0.575·11-s + 3.25·15-s − 0.813·17-s − 1.43·19-s − 2.03·21-s + 2.09·23-s + 2.88·25-s + 1.59·27-s + 2.68·29-s − 1.74·31-s + 0.775·33-s − 3.65·35-s + 0.342·37-s + 2.41·45-s + 1.10·47-s + 9/7·49-s − 1.09·51-s − 0.847·53-s + 1.39·55-s − 1.93·57-s − 1.34·59-s + 2.49·61-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

\(L(2)\) \(\approx\) \(4.661587197\)
\(L(\frac12)\) \(\approx\) \(4.661587197\)
\(L(\frac{5}{2})\) 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$C_2^2$ \( 1 - 7 T + 22 T^{2} - 7 p^{3} T^{3} + p^{6} T^{4} \)
5$C_2^2$ \( 1 - 27 T + 368 T^{2} - 27 p^{3} T^{3} + p^{6} T^{4} \)
11$C_2^2$ \( 1 - 21 T + 1478 T^{2} - 21 p^{3} T^{3} + p^{6} T^{4} \)
13$C_2^2$ \( 1 + 406 T^{2} + p^{6} T^{4} \)
17$C_2^2$ \( 1 + 57 T + 5996 T^{2} + 57 p^{3} T^{3} + p^{6} T^{4} \)
19$C_2^2$ \( 1 + 119 T + 7302 T^{2} + 119 p^{3} T^{3} + p^{6} T^{4} \)
23$C_2^2$ \( 1 - 231 T + 29954 T^{2} - 231 p^{3} T^{3} + p^{6} T^{4} \)
29$C_2$ \( ( 1 - 210 T + p^{3} T^{2} )^{2} \)
31$C_2^2$ \( 1 + 301 T + 60810 T^{2} + 301 p^{3} T^{3} + p^{6} T^{4} \)
37$C_2^2$ \( 1 - 77 T - 44724 T^{2} - 77 p^{3} T^{3} + p^{6} T^{4} \)
41$C_2^2$ \( 1 - 123970 T^{2} + p^{6} T^{4} \)
43$C_2^2$ \( 1 - 144314 T^{2} + p^{6} T^{4} \)
47$C_2^2$ \( 1 - 357 T + 23626 T^{2} - 357 p^{3} T^{3} + p^{6} T^{4} \)
53$C_2^2$ \( 1 + 327 T - 41948 T^{2} + 327 p^{3} T^{3} + p^{6} T^{4} \)
59$C_2^2$ \( 1 + 609 T + 165502 T^{2} + 609 p^{3} T^{3} + p^{6} T^{4} \)
61$C_2^2$ \( 1 - 1191 T + 699808 T^{2} - 1191 p^{3} T^{3} + p^{6} T^{4} \)
67$C_2^2$ \( 1 - 273 T + 325606 T^{2} - 273 p^{3} T^{3} + p^{6} T^{4} \)
71$C_2^2$ \( 1 - 668194 T^{2} + p^{6} T^{4} \)
73$C_2^2$ \( 1 - 99 T + 392284 T^{2} - 99 p^{3} T^{3} + p^{6} T^{4} \)
79$C_2^2$ \( 1 - 1407 T + 1152922 T^{2} - 1407 p^{3} T^{3} + p^{6} T^{4} \)
83$C_2$ \( ( 1 + 588 T + p^{3} T^{2} )^{2} \)
89$C_2^2$ \( 1 - 1275 T + 1246844 T^{2} - 1275 p^{3} T^{3} + p^{6} T^{4} \)
97$C_2^2$ \( 1 - 1728146 T^{2} + p^{6} T^{4} \)
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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

−13.53000078505154071868151626318, −13.07321191217106590846577290690, −12.62132310615714720702999035135, −12.31847069844987263460862416654, −10.85816125078438465822314012540, −10.68569976088432076823702288561, −10.03072161425810828489425285555, −9.449910207621869973488811424217, −9.261640121331326038067836595169, −8.797178039536995315865166572055, −8.318050664915791458055402638929, −6.98732471944585346244878936471, −6.55480910884229008419719043666, −6.43048809432552518270845036936, −5.46217136178364979887790938718, −4.65341426247453652243646408136, −3.59903378511289532832646633783, −2.54135708588691172172667420138, −2.53575801041562953237063997917, −1.23263943713534169703191204925, 1.23263943713534169703191204925, 2.53575801041562953237063997917, 2.54135708588691172172667420138, 3.59903378511289532832646633783, 4.65341426247453652243646408136, 5.46217136178364979887790938718, 6.43048809432552518270845036936, 6.55480910884229008419719043666, 6.98732471944585346244878936471, 8.318050664915791458055402638929, 8.797178039536995315865166572055, 9.261640121331326038067836595169, 9.449910207621869973488811424217, 10.03072161425810828489425285555, 10.68569976088432076823702288561, 10.85816125078438465822314012540, 12.31847069844987263460862416654, 12.62132310615714720702999035135, 13.07321191217106590846577290690, 13.53000078505154071868151626318

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