Properties

Label 4-3311e2-1.1-c0e2-0-1
Degree $4$
Conductor $10962721$
Sign $1$
Analytic cond. $2.73043$
Root an. cond. $1.28545$
Motivic weight $0$
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 + 4-s − 2·7-s − 2·8-s + 9-s − 2·11-s − 4·14-s − 4·16-s + 2·18-s − 4·22-s − 4·23-s + 25-s − 2·28-s − 2·29-s − 2·32-s + 36-s + 2·43-s − 2·44-s − 8·46-s + 3·49-s + 2·50-s − 2·53-s + 4·56-s − 4·58-s − 2·63-s + 3·64-s − 2·67-s + ⋯
L(s)  = 1  + 2·2-s + 4-s − 2·7-s − 2·8-s + 9-s − 2·11-s − 4·14-s − 4·16-s + 2·18-s − 4·22-s − 4·23-s + 25-s − 2·28-s − 2·29-s − 2·32-s + 36-s + 2·43-s − 2·44-s − 8·46-s + 3·49-s + 2·50-s − 2·53-s + 4·56-s − 4·58-s − 2·63-s + 3·64-s − 2·67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 10962721 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10962721 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(10962721\)    =    \(7^{2} \cdot 11^{2} \cdot 43^{2}\)
Sign: $1$
Analytic conductor: \(2.73043\)
Root analytic conductor: \(1.28545\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 10962721,\ (\ :0, 0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3874767902\)
\(L(\frac12)\) \(\approx\) \(0.3874767902\)
\(L(1)\) 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)$
bad7$C_1$ \( ( 1 + T )^{2} \)
11$C_1$ \( ( 1 + T )^{2} \)
43$C_1$ \( ( 1 - T )^{2} \)
good2$C_2$ \( ( 1 - T + T^{2} )^{2} \)
3$C_2^2$ \( 1 - T^{2} + T^{4} \)
5$C_2^2$ \( 1 - T^{2} + T^{4} \)
13$C_2$ \( ( 1 + T^{2} )^{2} \)
17$C_2^2$ \( 1 - T^{2} + T^{4} \)
19$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
23$C_1$ \( ( 1 + T )^{4} \)
29$C_2$ \( ( 1 + T + T^{2} )^{2} \)
31$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
37$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
41$C_2^2$ \( 1 - T^{2} + T^{4} \)
47$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
53$C_2$ \( ( 1 + T + T^{2} )^{2} \)
59$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
61$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
67$C_2$ \( ( 1 + T + T^{2} )^{2} \)
71$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
73$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
79$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
83$C_2^2$ \( 1 - T^{2} + T^{4} \)
89$C_2$ \( ( 1 + T^{2} )^{2} \)
97$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
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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

−8.950442625964970499725881326458, −8.782478952760063691003021097310, −8.136743777394137608595560109521, −7.75178268885156786753077014015, −7.38665007018077791666216631276, −7.11746300116069193607360004874, −6.26411059276351546498590140271, −6.21834174073620449732068242903, −5.84251601188439348149896939969, −5.77396384393688512860474928107, −5.03199486010886434058841200530, −4.85533111177461812743520028068, −4.22075374901353123559614346399, −3.84137852850043809542149813079, −3.83820096694568181422187927842, −3.23507497369810900200048463229, −2.60974997414010456890227051542, −2.58095675157051222116791334213, −1.81048874395588961463311634039, −0.24613183546964114335783284909, 0.24613183546964114335783284909, 1.81048874395588961463311634039, 2.58095675157051222116791334213, 2.60974997414010456890227051542, 3.23507497369810900200048463229, 3.83820096694568181422187927842, 3.84137852850043809542149813079, 4.22075374901353123559614346399, 4.85533111177461812743520028068, 5.03199486010886434058841200530, 5.77396384393688512860474928107, 5.84251601188439348149896939969, 6.21834174073620449732068242903, 6.26411059276351546498590140271, 7.11746300116069193607360004874, 7.38665007018077791666216631276, 7.75178268885156786753077014015, 8.136743777394137608595560109521, 8.782478952760063691003021097310, 8.950442625964970499725881326458

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