Properties

Label 4-112e2-1.1-c3e2-0-5
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  + 2·3-s + 22·5-s − 14·7-s + 6·9-s − 36·11-s + 42·13-s + 44·15-s + 8·17-s + 118·19-s − 28·21-s + 104·23-s + 170·25-s + 70·27-s − 56·29-s − 20·31-s − 72·33-s − 308·35-s + 504·37-s + 84·39-s − 544·41-s − 412·43-s + 132·45-s + 500·47-s + 147·49-s + 16·51-s + 268·53-s − 792·55-s + ⋯
L(s)  = 1  + 0.384·3-s + 1.96·5-s − 0.755·7-s + 2/9·9-s − 0.986·11-s + 0.896·13-s + 0.757·15-s + 0.114·17-s + 1.42·19-s − 0.290·21-s + 0.942·23-s + 1.35·25-s + 0.498·27-s − 0.358·29-s − 0.115·31-s − 0.379·33-s − 1.48·35-s + 2.23·37-s + 0.344·39-s − 2.07·41-s − 1.46·43-s + 0.437·45-s + 1.55·47-s + 3/7·49-s + 0.0439·51-s + 0.694·53-s − 1.94·55-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

\(L(2)\) \(\approx\) \(3.341561488\)
\(L(\frac12)\) \(\approx\) \(3.341561488\)
\(L(\frac{5}{2})\) 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$C_1$ \( ( 1 + p T )^{2} \)
good3$D_{4}$ \( 1 - 2 T - 2 T^{2} - 2 p^{3} T^{3} + p^{6} T^{4} \)
5$D_{4}$ \( 1 - 22 T + 314 T^{2} - 22 p^{3} T^{3} + p^{6} T^{4} \)
11$D_{4}$ \( 1 + 36 T + 934 T^{2} + 36 p^{3} T^{3} + p^{6} T^{4} \)
13$D_{4}$ \( 1 - 42 T + 3410 T^{2} - 42 p^{3} T^{3} + p^{6} T^{4} \)
17$D_{4}$ \( 1 - 8 T + 7790 T^{2} - 8 p^{3} T^{3} + p^{6} T^{4} \)
19$D_{4}$ \( 1 - 118 T + 7566 T^{2} - 118 p^{3} T^{3} + p^{6} T^{4} \)
23$D_{4}$ \( 1 - 104 T + 26126 T^{2} - 104 p^{3} T^{3} + p^{6} T^{4} \)
29$D_{4}$ \( 1 + 56 T + 21974 T^{2} + 56 p^{3} T^{3} + p^{6} T^{4} \)
31$D_{4}$ \( 1 + 20 T + 48510 T^{2} + 20 p^{3} T^{3} + p^{6} T^{4} \)
37$D_{4}$ \( 1 - 504 T + 159110 T^{2} - 504 p^{3} T^{3} + p^{6} T^{4} \)
41$D_{4}$ \( 1 + 544 T + 193358 T^{2} + 544 p^{3} T^{3} + p^{6} T^{4} \)
43$D_{4}$ \( 1 + 412 T + 190278 T^{2} + 412 p^{3} T^{3} + p^{6} T^{4} \)
47$D_{4}$ \( 1 - 500 T + 258974 T^{2} - 500 p^{3} T^{3} + p^{6} T^{4} \)
53$D_{4}$ \( 1 - 268 T + 20222 T^{2} - 268 p^{3} T^{3} + p^{6} T^{4} \)
59$D_{4}$ \( 1 - 198 T + 410926 T^{2} - 198 p^{3} T^{3} + p^{6} T^{4} \)
61$D_{4}$ \( 1 + 346 T + 429114 T^{2} + 346 p^{3} T^{3} + p^{6} T^{4} \)
67$D_{4}$ \( 1 + 1008 T + 591974 T^{2} + 1008 p^{3} T^{3} + p^{6} T^{4} \)
71$D_{4}$ \( 1 - 1224 T + 980014 T^{2} - 1224 p^{3} T^{3} + p^{6} T^{4} \)
73$D_{4}$ \( 1 - 716 T + 832326 T^{2} - 716 p^{3} T^{3} + p^{6} T^{4} \)
79$D_{4}$ \( 1 + 584 T + 997470 T^{2} + 584 p^{3} T^{3} + p^{6} T^{4} \)
83$D_{4}$ \( 1 - 1230 T + 1395886 T^{2} - 1230 p^{3} T^{3} + p^{6} T^{4} \)
89$D_{4}$ \( 1 - 596 T + 39542 T^{2} - 596 p^{3} T^{3} + p^{6} T^{4} \)
97$D_{4}$ \( 1 + 856 T + 1729230 T^{2} + 856 p^{3} T^{3} + p^{6} T^{4} \)
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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

−13.39183977160065018282084295939, −13.06820908011243355539440566482, −12.74603273755635636881703829792, −11.81423866386567139259177790670, −11.30953867050340791864034354295, −10.48158127029272947136481706747, −10.08133829424253986878913265825, −9.847779350536849921327799899203, −9.053156250372564023249049778649, −8.914141678411220504304682649405, −7.900891403867189870885225715046, −7.35786991238326427579520243825, −6.48817004331987790194434160734, −6.10839281101072543778073281045, −5.40481409386334407003653796655, −4.96655479380630335850166079436, −3.60889758676790384427048789961, −2.91073288803622947497691499124, −2.12925039107785246326992333153, −1.06124473172089153294017591543, 1.06124473172089153294017591543, 2.12925039107785246326992333153, 2.91073288803622947497691499124, 3.60889758676790384427048789961, 4.96655479380630335850166079436, 5.40481409386334407003653796655, 6.10839281101072543778073281045, 6.48817004331987790194434160734, 7.35786991238326427579520243825, 7.900891403867189870885225715046, 8.914141678411220504304682649405, 9.053156250372564023249049778649, 9.847779350536849921327799899203, 10.08133829424253986878913265825, 10.48158127029272947136481706747, 11.30953867050340791864034354295, 11.81423866386567139259177790670, 12.74603273755635636881703829792, 13.06820908011243355539440566482, 13.39183977160065018282084295939

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