Properties

Label 4-6048e2-1.1-c1e2-0-3
Degree $4$
Conductor $36578304$
Sign $1$
Analytic cond. $2332.26$
Root an. cond. $6.94935$
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  − 2·5-s + 2·7-s + 4·11-s + 6·17-s + 8·19-s − 8·23-s + 25-s − 8·31-s − 4·35-s − 6·37-s + 2·41-s + 10·43-s + 6·47-s + 3·49-s − 4·53-s − 8·55-s − 10·59-s + 12·61-s + 8·67-s − 8·71-s + 8·77-s + 2·79-s − 22·83-s − 12·85-s − 12·89-s − 16·95-s − 8·97-s + ⋯
L(s)  = 1  − 0.894·5-s + 0.755·7-s + 1.20·11-s + 1.45·17-s + 1.83·19-s − 1.66·23-s + 1/5·25-s − 1.43·31-s − 0.676·35-s − 0.986·37-s + 0.312·41-s + 1.52·43-s + 0.875·47-s + 3/7·49-s − 0.549·53-s − 1.07·55-s − 1.30·59-s + 1.53·61-s + 0.977·67-s − 0.949·71-s + 0.911·77-s + 0.225·79-s − 2.41·83-s − 1.30·85-s − 1.27·89-s − 1.64·95-s − 0.812·97-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 36578304 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36578304 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(36578304\)    =    \(2^{10} \cdot 3^{6} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(2332.26\)
Root analytic conductor: \(6.94935\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 36578304,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.986964266\)
\(L(\frac12)\) \(\approx\) \(2.986964266\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7$C_1$ \( ( 1 - T )^{2} \)
good5$D_{4}$ \( 1 + 2 T + 3 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
11$D_{4}$ \( 1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
13$C_2^2$ \( 1 + 18 T^{2} + p^{2} T^{4} \)
17$D_{4}$ \( 1 - 6 T + 35 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 - 8 T + 46 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
23$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
29$C_2^2$ \( 1 + 50 T^{2} + p^{2} T^{4} \)
31$D_{4}$ \( 1 + 8 T + 70 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 + 6 T + 51 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 - 2 T + 75 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 - 10 T + 103 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 - 6 T + 71 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
53$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
59$D_{4}$ \( 1 + 10 T + 111 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 - 12 T + 150 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
67$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
71$D_{4}$ \( 1 + 8 T + 126 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
73$C_2^2$ \( 1 + 114 T^{2} + p^{2} T^{4} \)
79$D_{4}$ \( 1 - 2 T + 151 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 + 22 T + 255 T^{2} + 22 p T^{3} + p^{2} T^{4} \)
89$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
97$D_{4}$ \( 1 + 8 T + 10 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
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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

−8.082296960076888904802630217763, −8.005536989658596281768854452131, −7.44551855835553024727235142303, −7.32243724935105443483460157591, −7.01317174147959844978427132310, −6.62188133774341958487151971261, −5.81874442639145192198258877998, −5.76350983504975032690343745459, −5.54269197240594627474143799295, −5.09120699371429884834835434206, −4.40720694398610572554533319256, −4.26252042643615297038216576122, −3.79283261301605203107269775773, −3.60431337725837845947277598668, −3.05786400232492525043737300232, −2.71897961403163818982807059730, −1.78683978135011886474510670710, −1.68318988369617434884633142313, −1.03821271148208423175406920275, −0.50791667873882477997765791007, 0.50791667873882477997765791007, 1.03821271148208423175406920275, 1.68318988369617434884633142313, 1.78683978135011886474510670710, 2.71897961403163818982807059730, 3.05786400232492525043737300232, 3.60431337725837845947277598668, 3.79283261301605203107269775773, 4.26252042643615297038216576122, 4.40720694398610572554533319256, 5.09120699371429884834835434206, 5.54269197240594627474143799295, 5.76350983504975032690343745459, 5.81874442639145192198258877998, 6.62188133774341958487151971261, 7.01317174147959844978427132310, 7.32243724935105443483460157591, 7.44551855835553024727235142303, 8.005536989658596281768854452131, 8.082296960076888904802630217763

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