Properties

Label 4-114e2-1.1-c1e2-0-8
Degree $4$
Conductor $12996$
Sign $1$
Analytic cond. $0.828636$
Root an. cond. $0.954093$
Motivic weight $1$
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  − 2·2-s + 3·3-s + 3·4-s − 6·6-s + 2·7-s − 4·8-s + 6·9-s + 9·12-s − 4·14-s + 5·16-s − 12·18-s − 8·19-s + 6·21-s − 12·24-s − 2·25-s + 9·27-s + 6·28-s − 18·29-s − 6·32-s + 18·36-s + 16·38-s − 12·42-s + 4·43-s + 15·48-s − 11·49-s + 4·50-s + 18·53-s + ⋯
L(s)  = 1  − 1.41·2-s + 1.73·3-s + 3/2·4-s − 2.44·6-s + 0.755·7-s − 1.41·8-s + 2·9-s + 2.59·12-s − 1.06·14-s + 5/4·16-s − 2.82·18-s − 1.83·19-s + 1.30·21-s − 2.44·24-s − 2/5·25-s + 1.73·27-s + 1.13·28-s − 3.34·29-s − 1.06·32-s + 3·36-s + 2.59·38-s − 1.85·42-s + 0.609·43-s + 2.16·48-s − 1.57·49-s + 0.565·50-s + 2.47·53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 12996 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12996 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(12996\)    =    \(2^{2} \cdot 3^{2} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(0.828636\)
Root analytic conductor: \(0.954093\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{114} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 12996,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.014876791\)
\(L(\frac12)\) \(\approx\) \(1.014876791\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 + T )^{2} \)
3$C_2$ \( 1 - p T + p T^{2} \)
19$C_2$ \( 1 + 8 T + p T^{2} \)
good5$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
7$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
11$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
17$C_2^2$ \( 1 - 31 T^{2} + p^{2} T^{4} \)
23$C_2^2$ \( 1 - 19 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + 9 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
37$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
41$C_2$ \( ( 1 + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 - 82 T^{2} + p^{2} T^{4} \)
53$C_2$ \( ( 1 - 9 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 - 59 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 11 T + p T^{2} )^{2} \)
79$C_2^2$ \( 1 - 110 T^{2} + p^{2} T^{4} \)
83$C_2^2$ \( 1 - 58 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
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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

−13.94405735128223976422455890320, −13.41443002050458089445793186250, −12.66236941589837179367898930518, −12.57579759618386388604901078259, −11.39982173113675917207642660543, −11.15459925579274283049481845686, −10.58443456365459907580287938768, −9.930035272462788786741653902004, −9.314696248225270502049248431688, −9.155950241897008087526607507583, −8.307453311780821382709945834705, −8.212552959238264029829531990035, −7.54327311687729818305855281059, −7.10187701398086923190915939762, −6.29901505145278870809570277864, −5.35039909813264016226809681016, −4.13413131291149390841434116951, −3.54725453331496676552181256947, −2.24970756960208927516047071219, −1.89648213860845155691770044950, 1.89648213860845155691770044950, 2.24970756960208927516047071219, 3.54725453331496676552181256947, 4.13413131291149390841434116951, 5.35039909813264016226809681016, 6.29901505145278870809570277864, 7.10187701398086923190915939762, 7.54327311687729818305855281059, 8.212552959238264029829531990035, 8.307453311780821382709945834705, 9.155950241897008087526607507583, 9.314696248225270502049248431688, 9.930035272462788786741653902004, 10.58443456365459907580287938768, 11.15459925579274283049481845686, 11.39982173113675917207642660543, 12.57579759618386388604901078259, 12.66236941589837179367898930518, 13.41443002050458089445793186250, 13.94405735128223976422455890320

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