Properties

Label 4-1216e2-1.1-c1e2-0-4
Degree $4$
Conductor $1478656$
Sign $1$
Analytic cond. $94.2803$
Root an. cond. $3.11605$
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  − 3-s + 3·5-s + 3·9-s − 8·11-s − 5·13-s − 3·15-s + 5·17-s + 8·19-s − 23-s + 5·25-s − 8·27-s + 3·29-s − 8·31-s + 8·33-s − 4·37-s + 5·39-s + 5·41-s + 11·43-s + 9·45-s − 5·47-s − 14·49-s − 5·51-s − 9·53-s − 24·55-s − 8·57-s − 13·59-s − 61-s + ⋯
L(s)  = 1  − 0.577·3-s + 1.34·5-s + 9-s − 2.41·11-s − 1.38·13-s − 0.774·15-s + 1.21·17-s + 1.83·19-s − 0.208·23-s + 25-s − 1.53·27-s + 0.557·29-s − 1.43·31-s + 1.39·33-s − 0.657·37-s + 0.800·39-s + 0.780·41-s + 1.67·43-s + 1.34·45-s − 0.729·47-s − 2·49-s − 0.700·51-s − 1.23·53-s − 3.23·55-s − 1.05·57-s − 1.69·59-s − 0.128·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1478656 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1478656 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1478656\)    =    \(2^{12} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(94.2803\)
Root analytic conductor: \(3.11605\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1216} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1478656,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.659727651\)
\(L(\frac12)\) \(\approx\) \(1.659727651\)
\(L(\frac{3}{2})\) 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 \( 1 \)
19$C_2$ \( 1 - 8 T + p T^{2} \)
good3$C_2^2$ \( 1 + T - 2 T^{2} + p T^{3} + p^{2} T^{4} \)
5$C_2^2$ \( 1 - 3 T + 4 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
7$C_2$ \( ( 1 + p T^{2} )^{2} \)
11$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
17$C_2^2$ \( 1 - 5 T + 8 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
23$C_2^2$ \( 1 + T - 22 T^{2} + p T^{3} + p^{2} T^{4} \)
29$C_2^2$ \( 1 - 3 T - 20 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
31$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 - 5 T - 16 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
43$C_2^2$ \( 1 - 11 T + 78 T^{2} - 11 p T^{3} + p^{2} T^{4} \)
47$C_2^2$ \( 1 + 5 T - 22 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
53$C_2^2$ \( 1 + 9 T + 28 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 + 13 T + 110 T^{2} + 13 p T^{3} + p^{2} T^{4} \)
61$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
67$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 11 T + p T^{2} ) \)
71$C_2^2$ \( 1 - T - 70 T^{2} - p T^{3} + p^{2} T^{4} \)
73$C_2^2$ \( 1 - 9 T + 8 T^{2} - 9 p T^{3} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 - 4 T + p T^{2} ) \)
83$C_2$ \( ( 1 - 16 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 + 3 T - 80 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
97$C_2^2$ \( 1 - 13 T + 72 T^{2} - 13 p T^{3} + p^{2} T^{4} \)
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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

−9.715180820042853961624359963375, −9.640916440371295828170286907160, −9.580556452842616133874707824549, −8.895873358642018484383126156128, −7.950334559990657000152670075033, −7.80740443640755218767881545153, −7.44493612495742036205885590942, −7.35394018323993776704115904559, −6.46276062270330181950929496883, −6.06109566245173621023143233127, −5.71773145201637802273942533246, −5.01657411371967860326931911384, −4.98646779021239085606869713691, −4.94281694520370834959707764673, −3.63621842988232333853591875633, −3.28109446348280584968007258376, −2.60821114431416979771409002139, −2.12525329649672895441203633233, −1.56284473853429568989784086135, −0.56279527096814167375525697871, 0.56279527096814167375525697871, 1.56284473853429568989784086135, 2.12525329649672895441203633233, 2.60821114431416979771409002139, 3.28109446348280584968007258376, 3.63621842988232333853591875633, 4.94281694520370834959707764673, 4.98646779021239085606869713691, 5.01657411371967860326931911384, 5.71773145201637802273942533246, 6.06109566245173621023143233127, 6.46276062270330181950929496883, 7.35394018323993776704115904559, 7.44493612495742036205885590942, 7.80740443640755218767881545153, 7.950334559990657000152670075033, 8.895873358642018484383126156128, 9.580556452842616133874707824549, 9.640916440371295828170286907160, 9.715180820042853961624359963375

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