Properties

Label 4-1014e2-1.1-c1e2-0-8
Degree $4$
Conductor $1028196$
Sign $1$
Analytic cond. $65.5586$
Root an. cond. $2.84549$
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  − 2-s − 3-s + 4·5-s + 6-s − 2·7-s + 8-s − 4·10-s + 2·14-s − 4·15-s − 16-s − 2·17-s + 6·19-s + 2·21-s + 4·23-s − 24-s + 2·25-s + 27-s + 10·29-s + 4·30-s + 20·31-s + 2·34-s − 8·35-s + 8·37-s − 6·38-s + 4·40-s − 10·41-s − 2·42-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 1.78·5-s + 0.408·6-s − 0.755·7-s + 0.353·8-s − 1.26·10-s + 0.534·14-s − 1.03·15-s − 1/4·16-s − 0.485·17-s + 1.37·19-s + 0.436·21-s + 0.834·23-s − 0.204·24-s + 2/5·25-s + 0.192·27-s + 1.85·29-s + 0.730·30-s + 3.59·31-s + 0.342·34-s − 1.35·35-s + 1.31·37-s − 0.973·38-s + 0.632·40-s − 1.56·41-s − 0.308·42-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1028196 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1028196 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1028196\)    =    \(2^{2} \cdot 3^{2} \cdot 13^{4}\)
Sign: $1$
Analytic conductor: \(65.5586\)
Root analytic conductor: \(2.84549\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1014} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1028196,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.735525755\)
\(L(\frac12)\) \(\approx\) \(1.735525755\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + T + T^{2} \)
3$C_2$ \( 1 + T + T^{2} \)
13 \( 1 \)
good5$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
7$C_2^2$ \( 1 + 2 T - 3 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
11$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
17$C_2^2$ \( 1 + 2 T - 13 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
19$C_2^2$ \( 1 - 6 T + 17 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
23$C_2^2$ \( 1 - 4 T - 7 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
29$C_2^2$ \( 1 - 10 T + 71 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
31$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
37$C_2^2$ \( 1 - 8 T + 27 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
41$C_2^2$ \( 1 + 10 T + 59 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
43$C_2^2$ \( 1 - 4 T - 27 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
47$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
53$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
59$C_2^2$ \( 1 - 4 T - 43 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 + 2 T - 57 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 - 2 T - 63 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
71$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
73$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 + p T^{2} )^{2} \)
83$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 + 6 T - 53 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
97$C_2^2$ \( 1 - 12 T + 47 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
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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

−9.943605321589757556460227478288, −9.916990580057499581559890464604, −9.443730558809914709056438519300, −9.099885430832980605017371436170, −8.524952657966552030407085928034, −8.360181206716990615801872853068, −7.52603363573901195948023823579, −7.31934016651896655492546429919, −6.51002941598703281298339055995, −6.39493597316988838302423067692, −6.01155636052248047055835378070, −5.62213911526220617674109292968, −4.90839654789549741958747741100, −4.73882444032061030566361779909, −4.02858383587444260502653930298, −3.14297483034777175511540972106, −2.59038320550320366500691043304, −2.32694789004965129489021697840, −1.11010681609343546837981652507, −0.891627287176597631210563176832, 0.891627287176597631210563176832, 1.11010681609343546837981652507, 2.32694789004965129489021697840, 2.59038320550320366500691043304, 3.14297483034777175511540972106, 4.02858383587444260502653930298, 4.73882444032061030566361779909, 4.90839654789549741958747741100, 5.62213911526220617674109292968, 6.01155636052248047055835378070, 6.39493597316988838302423067692, 6.51002941598703281298339055995, 7.31934016651896655492546429919, 7.52603363573901195948023823579, 8.360181206716990615801872853068, 8.524952657966552030407085928034, 9.099885430832980605017371436170, 9.443730558809914709056438519300, 9.916990580057499581559890464604, 9.943605321589757556460227478288

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