Properties

Label 4-927e2-1.1-c0e2-0-0
Degree $4$
Conductor $859329$
Sign $1$
Analytic cond. $0.214029$
Root an. cond. $0.680171$
Motivic weight $0$
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  + 2-s − 7-s − 13-s − 14-s + 17-s − 19-s + 23-s + 2·25-s − 26-s + 29-s − 32-s + 34-s − 38-s + 41-s + 46-s + 2·50-s + 58-s + 59-s − 61-s − 64-s − 79-s + 82-s + 83-s + 91-s − 97-s + 2·103-s − 4·107-s + ⋯
L(s)  = 1  + 2-s − 7-s − 13-s − 14-s + 17-s − 19-s + 23-s + 2·25-s − 26-s + 29-s − 32-s + 34-s − 38-s + 41-s + 46-s + 2·50-s + 58-s + 59-s − 61-s − 64-s − 79-s + 82-s + 83-s + 91-s − 97-s + 2·103-s − 4·107-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 859329 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 859329 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(859329\)    =    \(3^{4} \cdot 103^{2}\)
Sign: $1$
Analytic conductor: \(0.214029\)
Root analytic conductor: \(0.680171\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{927} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 859329,\ (\ :0, 0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.280651094\)
\(L(\frac12)\) \(\approx\) \(1.280651094\)
\(L(1)\) 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)$
bad3 \( 1 \)
103$C_1$ \( ( 1 - T )^{2} \)
good2$C_4$ \( 1 - T + T^{2} - T^{3} + T^{4} \)
5$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
7$C_4$ \( 1 + T + T^{2} + T^{3} + T^{4} \)
11$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
13$C_4$ \( 1 + T + T^{2} + T^{3} + T^{4} \)
17$C_4$ \( 1 - T + T^{2} - T^{3} + T^{4} \)
19$C_4$ \( 1 + T + T^{2} + T^{3} + T^{4} \)
23$C_4$ \( 1 - T + T^{2} - T^{3} + T^{4} \)
29$C_4$ \( 1 - T + T^{2} - T^{3} + T^{4} \)
31$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
37$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
41$C_4$ \( 1 - T + T^{2} - T^{3} + T^{4} \)
43$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
47$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
53$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
59$C_4$ \( 1 - T + T^{2} - T^{3} + T^{4} \)
61$C_4$ \( 1 + T + T^{2} + T^{3} + T^{4} \)
67$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
71$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
73$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
79$C_4$ \( 1 + T + T^{2} + T^{3} + T^{4} \)
83$C_4$ \( 1 - T + T^{2} - T^{3} + T^{4} \)
89$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
97$C_4$ \( 1 + T + T^{2} + T^{3} + T^{4} \)
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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

−10.77261549186022846070836068324, −10.04520232982095257651501639934, −9.560699256059565747450522245060, −9.453046778798123245451729413043, −8.708758436645538016550580608567, −8.555809977384639433432940183809, −7.932477664123689788445864580517, −7.32420320383045325708279180575, −6.94620883127531000143676040847, −6.74838133129326370058408346152, −6.06210862698941804644260382126, −5.68229934267069263904404091702, −5.07090324882585843755023687659, −4.74780865900708731254552737730, −4.39280252164025836403199604291, −3.79285988259939986749569744693, −3.02442426386372928453242059648, −2.97671953618363196033058754047, −2.13516838096298957613053787995, −1.01777354278783439579145493173, 1.01777354278783439579145493173, 2.13516838096298957613053787995, 2.97671953618363196033058754047, 3.02442426386372928453242059648, 3.79285988259939986749569744693, 4.39280252164025836403199604291, 4.74780865900708731254552737730, 5.07090324882585843755023687659, 5.68229934267069263904404091702, 6.06210862698941804644260382126, 6.74838133129326370058408346152, 6.94620883127531000143676040847, 7.32420320383045325708279180575, 7.932477664123689788445864580517, 8.555809977384639433432940183809, 8.708758436645538016550580608567, 9.453046778798123245451729413043, 9.560699256059565747450522245060, 10.04520232982095257651501639934, 10.77261549186022846070836068324

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