Properties

Label 4-48e4-1.1-c1e2-0-13
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 − 12·17-s + 8·23-s + 10·25-s + 20·31-s − 4·41-s + 24·47-s − 2·49-s − 8·71-s + 20·73-s − 12·79-s + 4·89-s − 12·97-s + 20·103-s + 28·113-s + 48·119-s + 6·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s − 32·161-s + 163-s + 167-s + ⋯
L(s)  = 1  − 1.51·7-s − 2.91·17-s + 1.66·23-s + 2·25-s + 3.59·31-s − 0.624·41-s + 3.50·47-s − 2/7·49-s − 0.949·71-s + 2.34·73-s − 1.35·79-s + 0.423·89-s − 1.21·97-s + 1.97·103-s + 2.63·113-s + 4.40·119-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 − 2.52·161-s + 0.0783·163-s + 0.0773·167-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

\(L(1)\) \(\approx\) \(1.917881343\)
\(L(\frac12)\) \(\approx\) \(1.917881343\)
\(L(\frac{3}{2})\) 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$C_2$ \( ( 1 - p T^{2} )^{2} \)
7$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
11$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
17$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 - p T^{2} )^{2} \)
23$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
31$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
41$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
47$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
53$C_2^2$ \( 1 + 38 T^{2} + p^{2} T^{4} \)
59$C_2^2$ \( 1 - 102 T^{2} + p^{2} T^{4} \)
61$C_2^2$ \( 1 - 118 T^{2} + p^{2} T^{4} \)
67$C_2^2$ \( 1 - 118 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
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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

−9.110811305845193024187413671908, −8.818442447272362450867433742920, −8.467787197629627861191951810615, −8.395300559329333244539484924573, −7.46618962569058683889751374971, −7.07818237263446246175161661843, −6.79656106667538285156952172592, −6.64029392073483037905432224881, −6.10083053128858053007161866083, −5.97295710103875823654212043959, −5.00550636684972483533667264911, −4.81069062198528587242616472805, −4.42424876166740720498172238122, −4.06396230468058290583962009579, −3.13160838501214896996928042036, −3.07563218131785714723678892280, −2.53484838795304958687526196288, −2.13107839483728461906007571724, −0.972974583802867190111204081286, −0.58923055414970614303377240373, 0.58923055414970614303377240373, 0.972974583802867190111204081286, 2.13107839483728461906007571724, 2.53484838795304958687526196288, 3.07563218131785714723678892280, 3.13160838501214896996928042036, 4.06396230468058290583962009579, 4.42424876166740720498172238122, 4.81069062198528587242616472805, 5.00550636684972483533667264911, 5.97295710103875823654212043959, 6.10083053128858053007161866083, 6.64029392073483037905432224881, 6.79656106667538285156952172592, 7.07818237263446246175161661843, 7.46618962569058683889751374971, 8.395300559329333244539484924573, 8.467787197629627861191951810615, 8.818442447272362450867433742920, 9.110811305845193024187413671908

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