Properties

Label 4-2e4-1.1-c6e2-0-0
Degree $4$
Conductor $16$
Sign $1$
Analytic cond. $0.846798$
Root an. cond. $0.959279$
Motivic weight $6$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 4·2-s − 48·4-s + 20·5-s − 448·8-s + 498·9-s + 80·10-s + 2.93e3·13-s + 1.28e3·16-s − 9.53e3·17-s + 1.99e3·18-s − 960·20-s − 3.09e4·25-s + 1.17e4·26-s + 5.09e4·29-s + 3.37e4·32-s − 3.81e4·34-s − 2.39e4·36-s + 3.98e3·37-s − 8.96e3·40-s + 5.87e4·41-s + 9.96e3·45-s + 1.39e5·49-s − 1.23e5·50-s − 1.40e5·52-s − 3.85e5·53-s + 2.03e5·58-s − 2.18e4·61-s + ⋯
L(s)  = 1  + 1/2·2-s − 3/4·4-s + 4/25·5-s − 7/8·8-s + 0.683·9-s + 2/25·10-s + 1.33·13-s + 5/16·16-s − 1.94·17-s + 0.341·18-s − 0.119·20-s − 1.98·25-s + 0.667·26-s + 2.09·29-s + 1.03·32-s − 0.970·34-s − 0.512·36-s + 0.0787·37-s − 0.139·40-s + 0.852·41-s + 0.109·45-s + 1.18·49-s − 0.990·50-s − 1.00·52-s − 2.59·53-s + 1.04·58-s − 0.0962·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+3)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(16\)    =    \(2^{4}\)
Sign: $1$
Analytic conductor: \(0.846798\)
Root analytic conductor: \(0.959279\)
Motivic weight: \(6\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{4} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 16,\ (\ :3, 3),\ 1)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(1.072695073\)
\(L(\frac12)\) \(\approx\) \(1.072695073\)
\(L(4)\) 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$C_2$ \( 1 - p^{2} T + p^{6} T^{2} \)
good3$C_2^2$ \( 1 - 166 p T^{2} + p^{12} T^{4} \)
5$C_2$ \( ( 1 - 2 p T + p^{6} T^{2} )^{2} \)
7$C_2^2$ \( 1 - 139298 T^{2} + p^{12} T^{4} \)
11$C_2^2$ \( 1 - 2620562 T^{2} + p^{12} T^{4} \)
13$C_2$ \( ( 1 - 1466 T + p^{6} T^{2} )^{2} \)
17$C_2$ \( ( 1 + 4766 T + p^{6} T^{2} )^{2} \)
19$C_2^2$ \( 1 - 37404722 T^{2} + p^{12} T^{4} \)
23$C_2^2$ \( 1 - 186397538 T^{2} + p^{12} T^{4} \)
29$C_2$ \( ( 1 - 25498 T + p^{6} T^{2} )^{2} \)
31$C_2^2$ \( 1 - 20219522 T^{2} + p^{12} T^{4} \)
37$C_2$ \( ( 1 - 1994 T + p^{6} T^{2} )^{2} \)
41$C_2$ \( ( 1 - 29362 T + p^{6} T^{2} )^{2} \)
43$C_2^2$ \( 1 - 12179022098 T^{2} + p^{12} T^{4} \)
47$C_2^2$ \( 1 - 21501276098 T^{2} + p^{12} T^{4} \)
53$C_2$ \( ( 1 + 192854 T + p^{6} T^{2} )^{2} \)
59$C_2^2$ \( 1 - 78211344722 T^{2} + p^{12} T^{4} \)
61$C_2$ \( ( 1 + 10918 T + p^{6} T^{2} )^{2} \)
67$C_2^2$ \( 1 - 25565876978 T^{2} + p^{12} T^{4} \)
71$C_2^2$ \( 1 + 27079768798 T^{2} + p^{12} T^{4} \)
73$C_2$ \( ( 1 - 288626 T + p^{6} T^{2} )^{2} \)
79$C_2^2$ \( 1 - 389636558402 T^{2} + p^{12} T^{4} \)
83$C_2^2$ \( 1 - 612202422578 T^{2} + p^{12} T^{4} \)
89$C_2$ \( ( 1 - 310738 T + p^{6} T^{2} )^{2} \)
97$C_2$ \( ( 1 + 1457086 T + p^{6} T^{2} )^{2} \)
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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

−24.53660273464275639743622462145, −23.57175299273089970011604651517, −23.26626467501858284215669449273, −22.19059552843461559609099182859, −21.61858622259975424842452197345, −20.95756433816106989269764042540, −19.83058320096091245143230269053, −18.90580774121245845310976603894, −17.88102778607635228745883699780, −17.59929793768923017837605476704, −15.87856099392868848643883166863, −15.43249017370527487601311095726, −13.89484124930942548130012896258, −13.47879660771318523930785751614, −12.41897331860520707654073360983, −11.03139954962341675621445453554, −9.604181779505845522978754643876, −8.405706884823255524905660661091, −6.27799477609798837982289869787, −4.29313366391969956478778604249, 4.29313366391969956478778604249, 6.27799477609798837982289869787, 8.405706884823255524905660661091, 9.604181779505845522978754643876, 11.03139954962341675621445453554, 12.41897331860520707654073360983, 13.47879660771318523930785751614, 13.89484124930942548130012896258, 15.43249017370527487601311095726, 15.87856099392868848643883166863, 17.59929793768923017837605476704, 17.88102778607635228745883699780, 18.90580774121245845310976603894, 19.83058320096091245143230269053, 20.95756433816106989269764042540, 21.61858622259975424842452197345, 22.19059552843461559609099182859, 23.26626467501858284215669449273, 23.57175299273089970011604651517, 24.53660273464275639743622462145

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