Properties

Label 4-1216e2-1.1-c1e2-0-19
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  + 6·7-s + 5·9-s + 6·17-s − 18·23-s + 10·25-s + 12·31-s + 12·41-s + 13·49-s + 30·63-s + 22·73-s − 24·79-s + 16·81-s − 16·97-s + 12·103-s − 24·113-s + 36·119-s + 22·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 30·153-s + 157-s − 108·161-s + 163-s + ⋯
L(s)  = 1  + 2.26·7-s + 5/3·9-s + 1.45·17-s − 3.75·23-s + 2·25-s + 2.15·31-s + 1.87·41-s + 13/7·49-s + 3.77·63-s + 2.57·73-s − 2.70·79-s + 16/9·81-s − 1.62·97-s + 1.18·103-s − 2.25·113-s + 3.30·119-s + 2·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 2.42·153-s + 0.0798·157-s − 8.51·161-s + 0.0783·163-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

\(L(1)\) \(\approx\) \(4.097761469\)
\(L(\frac12)\) \(\approx\) \(4.097761469\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + T^{2} \)
good3$C_2^2$ \( 1 - 5 T^{2} + p^{2} T^{4} \)
5$C_2$ \( ( 1 - p T^{2} )^{2} \)
7$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
11$C_2$ \( ( 1 - p T^{2} )^{2} \)
13$C_2^2$ \( 1 - 17 T^{2} + p^{2} T^{4} \)
17$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 + 9 T + p T^{2} )^{2} \)
29$C_2^2$ \( 1 + 23 T^{2} + p^{2} T^{4} \)
31$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
37$C_2^2$ \( 1 - 38 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
47$C_2$ \( ( 1 + p T^{2} )^{2} \)
53$C_2^2$ \( 1 - 25 T^{2} + p^{2} T^{4} \)
59$C_2^2$ \( 1 - 109 T^{2} + p^{2} T^{4} \)
61$C_2^2$ \( 1 - 86 T^{2} + p^{2} T^{4} \)
67$C_2^2$ \( 1 - 109 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 11 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 + p T^{2} )^{2} \)
97$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
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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

−9.862685587425394053191256341060, −9.809603195376979946669488949759, −9.145816530441427860233828752262, −8.462742243307877688021409433914, −8.105497545734099352261462033095, −8.040319249155155768203095438866, −7.58651326697501713051468396478, −7.28668043003824428231687611588, −6.46359765609492926333934285601, −6.37254600139472960295505130365, −5.51248430504078833539322710571, −5.35132135944355142848853256403, −4.57082014948238083712548335703, −4.39078225807291656867764951514, −4.14829853802996598658054914664, −3.40579877635572868078992330615, −2.52003549943530161976396997584, −2.04295540264713895441864435695, −1.32624340253520185142858625608, −1.07113267627942141792377710412, 1.07113267627942141792377710412, 1.32624340253520185142858625608, 2.04295540264713895441864435695, 2.52003549943530161976396997584, 3.40579877635572868078992330615, 4.14829853802996598658054914664, 4.39078225807291656867764951514, 4.57082014948238083712548335703, 5.35132135944355142848853256403, 5.51248430504078833539322710571, 6.37254600139472960295505130365, 6.46359765609492926333934285601, 7.28668043003824428231687611588, 7.58651326697501713051468396478, 8.040319249155155768203095438866, 8.105497545734099352261462033095, 8.462742243307877688021409433914, 9.145816530441427860233828752262, 9.809603195376979946669488949759, 9.862685587425394053191256341060

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