Properties

Label 4-48e4-1.1-c1e2-0-24
Degree $4$
Conductor $5308416$
Sign $1$
Analytic cond. $338.469$
Root an. cond. $4.28923$
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  + 4·7-s − 8·17-s + 16·23-s + 6·25-s + 12·31-s + 24·41-s − 16·47-s − 2·49-s − 4·73-s + 28·79-s − 16·89-s − 4·97-s + 28·103-s − 16·113-s − 32·119-s + 6·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 64·161-s + 163-s + 167-s + 22·169-s + ⋯
L(s)  = 1  + 1.51·7-s − 1.94·17-s + 3.33·23-s + 6/5·25-s + 2.15·31-s + 3.74·41-s − 2.33·47-s − 2/7·49-s − 0.468·73-s + 3.15·79-s − 1.69·89-s − 0.406·97-s + 2.75·103-s − 1.50·113-s − 2.93·119-s + 6/11·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 5.04·161-s + 0.0783·163-s + 0.0773·167-s + 1.69·169-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5308416 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5308416 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(5308416\)    =    \(2^{16} \cdot 3^{4}\)
Sign: $1$
Analytic conductor: \(338.469\)
Root analytic conductor: \(4.28923\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{2304} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 5308416,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.818959941\)
\(L(\frac12)\) \(\approx\) \(3.818959941\)
\(L(\frac{3}{2})\) 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 \( 1 \)
good5$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
7$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
11$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \)
13$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
17$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
19$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
23$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
29$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
31$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
41$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 + 58 T^{2} + p^{2} T^{4} \)
47$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
53$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
59$C_2^2$ \( 1 - 54 T^{2} + p^{2} T^{4} \)
61$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
67$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
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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

−9.071770219641167348781814342603, −8.911966787337030074926601960844, −8.291075298961237783859183623115, −8.218781264440375284238641931958, −7.70541239741409786635088172892, −7.28633607395926117884851400993, −6.81602000442399146212800856492, −6.52753032453984109426690266289, −6.29062222874349048401808275412, −5.50937409345539298850416806282, −5.07877775972606713572339068965, −4.75462772857192464273749896097, −4.47799024983447709974739020709, −4.28937179762605532913910805486, −3.21736699297368596350936873975, −2.98823899997234689004799977554, −2.44386756708720472514946110427, −1.92630733709849888654509153709, −1.10692434359277039338905228255, −0.814333099439137324094436879149, 0.814333099439137324094436879149, 1.10692434359277039338905228255, 1.92630733709849888654509153709, 2.44386756708720472514946110427, 2.98823899997234689004799977554, 3.21736699297368596350936873975, 4.28937179762605532913910805486, 4.47799024983447709974739020709, 4.75462772857192464273749896097, 5.07877775972606713572339068965, 5.50937409345539298850416806282, 6.29062222874349048401808275412, 6.52753032453984109426690266289, 6.81602000442399146212800856492, 7.28633607395926117884851400993, 7.70541239741409786635088172892, 8.218781264440375284238641931958, 8.291075298961237783859183623115, 8.911966787337030074926601960844, 9.071770219641167348781814342603

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