Properties

Label 4-1386e2-1.1-c1e2-0-2
Degree $4$
Conductor $1920996$
Sign $1$
Analytic cond. $122.484$
Root an. cond. $3.32675$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 4-s + 4·7-s − 3·8-s − 4·11-s + 4·14-s − 16-s − 4·22-s − 4·23-s + 2·25-s − 4·28-s − 6·29-s + 5·32-s + 8·37-s − 12·43-s + 4·44-s − 4·46-s + 9·49-s + 2·50-s − 12·56-s − 6·58-s + 7·64-s + 12·67-s − 20·71-s + 8·74-s − 16·77-s + 16·79-s + ⋯
L(s)  = 1  + 0.707·2-s − 1/2·4-s + 1.51·7-s − 1.06·8-s − 1.20·11-s + 1.06·14-s − 1/4·16-s − 0.852·22-s − 0.834·23-s + 2/5·25-s − 0.755·28-s − 1.11·29-s + 0.883·32-s + 1.31·37-s − 1.82·43-s + 0.603·44-s − 0.589·46-s + 9/7·49-s + 0.282·50-s − 1.60·56-s − 0.787·58-s + 7/8·64-s + 1.46·67-s − 2.37·71-s + 0.929·74-s − 1.82·77-s + 1.80·79-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1920996 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1920996 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1920996\)    =    \(2^{2} \cdot 3^{4} \cdot 7^{2} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(122.484\)
Root analytic conductor: \(3.32675\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1920996,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.060409758\)
\(L(\frac12)\) \(\approx\) \(2.060409758\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - T + p T^{2} \)
3 \( 1 \)
7$C_2$ \( 1 - 4 T + p T^{2} \)
11$C_2$ \( 1 + 4 T + p T^{2} \)
good5$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
13$C_2^2$ \( 1 - 16 T^{2} + p^{2} T^{4} \)
17$C_2^2$ \( 1 - 16 T^{2} + p^{2} T^{4} \)
19$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \)
23$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
29$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
31$C_2^2$ \( 1 + 48 T^{2} + p^{2} T^{4} \)
37$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \)
41$C_2^2$ \( 1 - 32 T^{2} + p^{2} T^{4} \)
43$C_2$$\times$$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
47$C_2^2$ \( 1 + 32 T^{2} + p^{2} T^{4} \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
59$C_2^2$ \( 1 + 80 T^{2} + p^{2} T^{4} \)
61$C_2^2$ \( 1 + 64 T^{2} + p^{2} T^{4} \)
67$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 - 4 T + p T^{2} ) \)
71$C_2$$\times$$C_2$ \( ( 1 + 8 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
73$C_2^2$ \( 1 - 16 T^{2} + p^{2} T^{4} \)
79$C_2$$\times$$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + p T^{2} ) \)
83$C_2^2$ \( 1 + 46 T^{2} + p^{2} T^{4} \)
89$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
97$C_2^2$ \( 1 + 98 T^{2} + p^{2} T^{4} \)
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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

−7.80710234986750295215171266900, −7.51681118112331058704313459168, −6.93771207290685874327252071806, −6.30956904237721901273888150975, −5.96078402913610306128312129098, −5.39183712065863038997508616360, −5.11687754541231297484395286951, −4.87190106526210727490718789783, −4.20706628176363598110907978154, −4.00107710044546481657966048650, −3.25428933242151959371053646494, −2.76964787297344771221679855972, −2.12276993024608164469907480830, −1.57634690860953738285002084526, −0.52327886666068684536004689873, 0.52327886666068684536004689873, 1.57634690860953738285002084526, 2.12276993024608164469907480830, 2.76964787297344771221679855972, 3.25428933242151959371053646494, 4.00107710044546481657966048650, 4.20706628176363598110907978154, 4.87190106526210727490718789783, 5.11687754541231297484395286951, 5.39183712065863038997508616360, 5.96078402913610306128312129098, 6.30956904237721901273888150975, 6.93771207290685874327252071806, 7.51681118112331058704313459168, 7.80710234986750295215171266900

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