Properties

Label 4-3240e2-1.1-c1e2-0-3
Degree $4$
Conductor $10497600$
Sign $1$
Analytic cond. $669.336$
Root an. cond. $5.08640$
Motivic weight $1$
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  + 5-s − 11-s − 14·19-s − 6·23-s + 7·29-s − 31-s − 4·37-s − 9·41-s + 6·43-s + 2·47-s + 7·49-s − 55-s − 3·59-s + 10·61-s + 2·67-s + 2·71-s − 4·79-s + 6·83-s − 14·89-s − 14·95-s − 2·97-s + 9·101-s + 6·103-s − 4·107-s + 6·109-s − 6·115-s + 11·121-s + ⋯
L(s)  = 1  + 0.447·5-s − 0.301·11-s − 3.21·19-s − 1.25·23-s + 1.29·29-s − 0.179·31-s − 0.657·37-s − 1.40·41-s + 0.914·43-s + 0.291·47-s + 49-s − 0.134·55-s − 0.390·59-s + 1.28·61-s + 0.244·67-s + 0.237·71-s − 0.450·79-s + 0.658·83-s − 1.48·89-s − 1.43·95-s − 0.203·97-s + 0.895·101-s + 0.591·103-s − 0.386·107-s + 0.574·109-s − 0.559·115-s + 121-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 10497600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10497600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(10497600\)    =    \(2^{6} \cdot 3^{8} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(669.336\)
Root analytic conductor: \(5.08640\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{3240} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 10497600,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9871190536\)
\(L(\frac12)\) \(\approx\) \(0.9871190536\)
\(L(\frac{3}{2})\) 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 \( 1 \)
3 \( 1 \)
5$C_2$ \( 1 - T + T^{2} \)
good7$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
11$C_2^2$ \( 1 + T - 10 T^{2} + p T^{3} + p^{2} T^{4} \)
13$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
17$C_2$ \( ( 1 + p T^{2} )^{2} \)
19$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 + 6 T + 13 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
29$C_2^2$ \( 1 - 7 T + 20 T^{2} - 7 p T^{3} + p^{2} T^{4} \)
31$C_2^2$ \( 1 + T - 30 T^{2} + p T^{3} + p^{2} T^{4} \)
37$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 + 9 T + 40 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
43$C_2^2$ \( 1 - 6 T - 7 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 2 T - 43 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
53$C_2$ \( ( 1 + p T^{2} )^{2} \)
59$C_2^2$ \( 1 + 3 T - 50 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 - 10 T + 39 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 - 2 T - 63 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 + 17 T + p T^{2} ) \)
83$C_2^2$ \( 1 - 6 T - 47 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
89$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \)
97$C_2^2$ \( 1 + 2 T - 93 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
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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

−8.705550847400631241583998353425, −8.454310084063956440900752753337, −8.311867967741241590190328438237, −7.75607075380323243652641582952, −7.38221592576758449722141548656, −6.80291345804993444093805886201, −6.54794710929142880842536755335, −6.31727944437505973205507161240, −5.85927333511413202358990212732, −5.47316120474614583502508455091, −5.03952161400581239660405913048, −4.46591148130646932604413443463, −4.25481758404624996939780322094, −3.83428637614156016290375641886, −3.33419829508279863597697056492, −2.58169819546884810541221990767, −2.26887868781461605085451918475, −1.97793032664194551512798976238, −1.25285706293851046893731787580, −0.30101653910597770879682821785, 0.30101653910597770879682821785, 1.25285706293851046893731787580, 1.97793032664194551512798976238, 2.26887868781461605085451918475, 2.58169819546884810541221990767, 3.33419829508279863597697056492, 3.83428637614156016290375641886, 4.25481758404624996939780322094, 4.46591148130646932604413443463, 5.03952161400581239660405913048, 5.47316120474614583502508455091, 5.85927333511413202358990212732, 6.31727944437505973205507161240, 6.54794710929142880842536755335, 6.80291345804993444093805886201, 7.38221592576758449722141548656, 7.75607075380323243652641582952, 8.311867967741241590190328438237, 8.454310084063956440900752753337, 8.705550847400631241583998353425

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