Properties

Label 4-2347884-1.1-c1e2-0-14
Degree $4$
Conductor $2347884$
Sign $1$
Analytic cond. $149.703$
Root an. cond. $3.49790$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s + 4-s − 4·5-s + 3·9-s + 11-s − 2·12-s + 8·15-s + 16-s − 4·20-s − 8·23-s + 2·25-s − 4·27-s − 8·31-s − 2·33-s + 3·36-s − 4·37-s + 44-s − 12·45-s − 16·47-s − 2·48-s + 49-s − 28·53-s − 4·55-s + 24·59-s + 8·60-s + 64-s + 8·67-s + ⋯
L(s)  = 1  − 1.15·3-s + 1/2·4-s − 1.78·5-s + 9-s + 0.301·11-s − 0.577·12-s + 2.06·15-s + 1/4·16-s − 0.894·20-s − 1.66·23-s + 2/5·25-s − 0.769·27-s − 1.43·31-s − 0.348·33-s + 1/2·36-s − 0.657·37-s + 0.150·44-s − 1.78·45-s − 2.33·47-s − 0.288·48-s + 1/7·49-s − 3.84·53-s − 0.539·55-s + 3.12·59-s + 1.03·60-s + 1/8·64-s + 0.977·67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2347884 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2347884 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(2347884\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2} \cdot 11^{3}\)
Sign: $1$
Analytic conductor: \(149.703\)
Root analytic conductor: \(3.49790\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{2347884} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 2347884,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
3$C_1$ \( ( 1 + T )^{2} \)
7$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
11$C_1$ \( 1 - T \)
good5$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
17$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
19$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
23$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
31$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
43$C_2$ \( ( 1 + p T^{2} )^{2} \)
47$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
53$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
67$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
79$C_2$ \( ( 1 + p T^{2} )^{2} \)
83$C_2$ \( ( 1 + p T^{2} )^{2} \)
89$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \)
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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

−7.13188734812014929646560473693, −6.91394332771191655575545668718, −6.36362927444081124983699082758, −6.17569848665337782312463950999, −5.47060768957514705173421460731, −5.12658951935756468823062559161, −4.72832473817225769292057457440, −3.95586600708170815846951891453, −3.72716170584453900058590617219, −3.62319053698876325772393238292, −2.60592425173918161360361657651, −1.87596574858564202101556695192, −1.31751202956692819023526147888, 0, 0, 1.31751202956692819023526147888, 1.87596574858564202101556695192, 2.60592425173918161360361657651, 3.62319053698876325772393238292, 3.72716170584453900058590617219, 3.95586600708170815846951891453, 4.72832473817225769292057457440, 5.12658951935756468823062559161, 5.47060768957514705173421460731, 6.17569848665337782312463950999, 6.36362927444081124983699082758, 6.91394332771191655575545668718, 7.13188734812014929646560473693

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