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Properties

Label 4-1372-1.1-c1e2-0-0
Degree $4$
Conductor $1372$
Sign $1$
Analytic cond. $0.0874799$
Root an. cond. $0.543847$
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 + 3·4-s + 7-s − 4·8-s − 2·9-s − 2·14-s + 5·16-s + 4·18-s − 10·25-s + 3·28-s − 12·29-s − 6·32-s − 6·36-s + 4·37-s + 16·43-s + 49-s + 20·50-s + 12·53-s − 4·56-s + 24·58-s − 2·63-s + 7·64-s − 8·67-s + 8·72-s − 8·74-s + 16·79-s − 5·81-s + ⋯
L(s)  = 1  − 1.41·2-s + 3/2·4-s + 0.377·7-s − 1.41·8-s − 2/3·9-s − 0.534·14-s + 5/4·16-s + 0.942·18-s − 2·25-s + 0.566·28-s − 2.22·29-s − 1.06·32-s − 36-s + 0.657·37-s + 2.43·43-s + 1/7·49-s + 2.82·50-s + 1.64·53-s − 0.534·56-s + 3.15·58-s − 0.251·63-s + 7/8·64-s − 0.977·67-s + 0.942·72-s − 0.929·74-s + 1.80·79-s − 5/9·81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1372 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1372 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1372\)    =    \(2^{2} \cdot 7^{3}\)
Sign: $1$
Analytic conductor: \(0.0874799\)
Root analytic conductor: \(0.543847\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1372,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3308765761\)
\(L(\frac12)\) \(\approx\) \(0.3308765761\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 + T )^{2} \)
7$C_1$ \( 1 - T \)
good3$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
5$C_2$ \( ( 1 + p T^{2} )^{2} \)
11$C_2$ \( ( 1 + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
17$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
19$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
23$C_2$ \( ( 1 + p T^{2} )^{2} \)
29$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
37$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
43$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
61$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
67$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
89$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
97$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
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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

−13.87946529878520861319815819980, −13.15728760628296679290378410395, −12.30526099005271094386573698884, −11.59222775413431567626281153330, −11.23136141438460806904855801814, −10.57696390398788315518055643332, −9.765547119459919407856461234632, −9.205467104844532229657218112988, −8.624667079434487291694904422005, −7.57571100088867902110310233811, −7.55563465371540871894331307120, −6.12986038892916091786195943044, −5.57928681742950427486583645839, −3.87985229263682105648190826671, −2.25140513775369021989127014661, 2.25140513775369021989127014661, 3.87985229263682105648190826671, 5.57928681742950427486583645839, 6.12986038892916091786195943044, 7.55563465371540871894331307120, 7.57571100088867902110310233811, 8.624667079434487291694904422005, 9.205467104844532229657218112988, 9.765547119459919407856461234632, 10.57696390398788315518055643332, 11.23136141438460806904855801814, 11.59222775413431567626281153330, 12.30526099005271094386573698884, 13.15728760628296679290378410395, 13.87946529878520861319815819980

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