Properties

Label 4-2736e2-1.1-c1e2-0-26
Degree $4$
Conductor $7485696$
Sign $1$
Analytic cond. $477.294$
Root an. cond. $4.67408$
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  − 4·5-s + 6·11-s − 2·13-s + 2·17-s − 19-s − 6·23-s + 5·25-s − 4·29-s + 20·31-s + 4·37-s + 9·41-s − 4·43-s + 12·47-s − 14·49-s − 2·53-s − 24·55-s + 59-s + 8·61-s + 8·65-s + 9·67-s + 6·71-s + 9·73-s − 4·79-s − 10·83-s − 8·85-s − 18·89-s + 4·95-s + ⋯
L(s)  = 1  − 1.78·5-s + 1.80·11-s − 0.554·13-s + 0.485·17-s − 0.229·19-s − 1.25·23-s + 25-s − 0.742·29-s + 3.59·31-s + 0.657·37-s + 1.40·41-s − 0.609·43-s + 1.75·47-s − 2·49-s − 0.274·53-s − 3.23·55-s + 0.130·59-s + 1.02·61-s + 0.992·65-s + 1.09·67-s + 0.712·71-s + 1.05·73-s − 0.450·79-s − 1.09·83-s − 0.867·85-s − 1.90·89-s + 0.410·95-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7485696 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7485696 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(7485696\)    =    \(2^{8} \cdot 3^{4} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(477.294\)
Root analytic conductor: \(4.67408\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{2736} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 7485696,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.882376994\)
\(L(\frac12)\) \(\approx\) \(1.882376994\)
\(L(\frac{3}{2})\) 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 \( 1 \)
19$C_2$ \( 1 + T + p T^{2} \)
good5$C_2^2$ \( 1 + 4 T + 11 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
7$C_2$ \( ( 1 + p T^{2} )^{2} \)
11$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
17$C_2^2$ \( 1 - 2 T - 13 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
23$C_2^2$ \( 1 + 6 T + 13 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
29$C_2^2$ \( 1 + 4 T - 13 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
31$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
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$C_2^2$ \( 1 + 4 T - 27 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 12 T + 97 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
53$C_2^2$ \( 1 + 2 T - 49 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 - T - 58 T^{2} - p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 - 8 T + 3 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 - 9 T + 14 T^{2} - 9 p T^{3} + p^{2} T^{4} \)
71$C_2^2$ \( 1 - 6 T - 35 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
73$C_2^2$ \( 1 - 9 T + 8 T^{2} - 9 p T^{3} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 + 17 T + p T^{2} ) \)
83$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 + 18 T + 235 T^{2} + 18 p T^{3} + p^{2} T^{4} \)
97$C_2^2$ \( 1 + T - 96 T^{2} + p T^{3} + p^{2} T^{4} \)
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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

−8.696197743870196792576350317335, −8.691218851785287854061457539685, −8.201785036080650947522854094004, −7.87804827729637453583934437159, −7.55862720882922425588903274927, −7.29290000988292650227785123127, −6.55552720790682901350491756988, −6.54690533405442325656730210136, −6.04531976640512516290977241350, −5.66179233959072772058893857353, −4.82734747068053733326938221757, −4.63561350299984874636273359287, −4.08979577855155459500776413082, −4.00013051104788343108469599350, −3.52349742961524866540511510952, −3.00625379657262912209151224413, −2.43073321518822523297544650794, −1.81986019631388670963194640673, −0.957658331862954592834736478013, −0.57557960081312173414355127123, 0.57557960081312173414355127123, 0.957658331862954592834736478013, 1.81986019631388670963194640673, 2.43073321518822523297544650794, 3.00625379657262912209151224413, 3.52349742961524866540511510952, 4.00013051104788343108469599350, 4.08979577855155459500776413082, 4.63561350299984874636273359287, 4.82734747068053733326938221757, 5.66179233959072772058893857353, 6.04531976640512516290977241350, 6.54690533405442325656730210136, 6.55552720790682901350491756988, 7.29290000988292650227785123127, 7.55862720882922425588903274927, 7.87804827729637453583934437159, 8.201785036080650947522854094004, 8.691218851785287854061457539685, 8.696197743870196792576350317335

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