Properties

Label 4-4598e2-1.1-c1e2-0-2
Degree $4$
Conductor $21141604$
Sign $1$
Analytic cond. $1348.00$
Root an. cond. $6.05930$
Motivic weight $1$
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  − 2·2-s − 2·3-s + 3·4-s + 2·5-s + 4·6-s + 4·7-s − 4·8-s − 3·9-s − 4·10-s − 6·12-s − 8·14-s − 4·15-s + 5·16-s − 8·17-s + 6·18-s + 2·19-s + 6·20-s − 8·21-s − 2·23-s + 8·24-s − 4·25-s + 14·27-s + 12·28-s + 8·30-s + 4·31-s − 6·32-s + 16·34-s + ⋯
L(s)  = 1  − 1.41·2-s − 1.15·3-s + 3/2·4-s + 0.894·5-s + 1.63·6-s + 1.51·7-s − 1.41·8-s − 9-s − 1.26·10-s − 1.73·12-s − 2.13·14-s − 1.03·15-s + 5/4·16-s − 1.94·17-s + 1.41·18-s + 0.458·19-s + 1.34·20-s − 1.74·21-s − 0.417·23-s + 1.63·24-s − 4/5·25-s + 2.69·27-s + 2.26·28-s + 1.46·30-s + 0.718·31-s − 1.06·32-s + 2.74·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 21141604 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21141604 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(21141604\)    =    \(2^{2} \cdot 11^{4} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(1348.00\)
Root analytic conductor: \(6.05930\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{4598} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 21141604,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5862918007\)
\(L(\frac12)\) \(\approx\) \(0.5862918007\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 + T )^{2} \)
11 \( 1 \)
19$C_1$ \( ( 1 - T )^{2} \)
good3$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
5$D_{4}$ \( 1 - 2 T + 8 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
7$D_{4}$ \( 1 - 4 T + 15 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
13$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \)
17$D_{4}$ \( 1 + 8 T + 38 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
23$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
29$C_2^2$ \( 1 + 55 T^{2} + p^{2} T^{4} \)
31$D_{4}$ \( 1 - 4 T + 54 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 + 6 T + 71 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 + 18 T + 160 T^{2} + 18 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 + 4 T + 42 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 - 18 T + 163 T^{2} - 18 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 + 6 T + 67 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 - 2 T + 11 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 - 6 T + 128 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
67$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \)
71$D_{4}$ \( 1 - 18 T + 196 T^{2} - 18 p T^{3} + p^{2} T^{4} \)
73$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
79$D_{4}$ \( 1 - 6 T + 140 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 + 18 T + 220 T^{2} + 18 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 - 14 T + 200 T^{2} - 14 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 - 20 T + 282 T^{2} - 20 p T^{3} + p^{2} T^{4} \)
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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

−8.566396875021535060429392152187, −8.321097595735992382980196174602, −7.77227666676305595634903426783, −7.68840621609576732569336126168, −6.82272702198640737011776094892, −6.72618362907029438530955267240, −6.37262799311460060404166761712, −6.18145430533075254112641389814, −5.40494735038357138726525029594, −5.38266803331276432754755090571, −5.01021047976185066630510244435, −4.76852537712280093870554148054, −3.74924779892313023992806743840, −3.73179439196756753725609952680, −2.66527301920239186644644347413, −2.49165185844933903023332061351, −1.87687307597278187780052376503, −1.76092782896980712459720622103, −0.887839607975114701993999935570, −0.34922137815240737037392207977, 0.34922137815240737037392207977, 0.887839607975114701993999935570, 1.76092782896980712459720622103, 1.87687307597278187780052376503, 2.49165185844933903023332061351, 2.66527301920239186644644347413, 3.73179439196756753725609952680, 3.74924779892313023992806743840, 4.76852537712280093870554148054, 5.01021047976185066630510244435, 5.38266803331276432754755090571, 5.40494735038357138726525029594, 6.18145430533075254112641389814, 6.37262799311460060404166761712, 6.72618362907029438530955267240, 6.82272702198640737011776094892, 7.68840621609576732569336126168, 7.77227666676305595634903426783, 8.321097595735992382980196174602, 8.566396875021535060429392152187

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