Properties

Label 4-2347884-1.1-c1e2-0-7
Degree $4$
Conductor $2347884$
Sign $-1$
Analytic cond. $149.703$
Root an. cond. $3.49790$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s − 4-s + 2·5-s + 3·9-s + 11-s + 2·12-s − 4·15-s + 16-s − 2·20-s + 2·23-s − 6·25-s − 4·27-s − 8·31-s − 2·33-s − 3·36-s + 4·37-s − 44-s + 6·45-s − 2·47-s − 2·48-s + 49-s − 2·53-s + 2·55-s + 4·60-s − 64-s − 4·69-s + 10·71-s + ⋯
L(s)  = 1  − 1.15·3-s − 1/2·4-s + 0.894·5-s + 9-s + 0.301·11-s + 0.577·12-s − 1.03·15-s + 1/4·16-s − 0.447·20-s + 0.417·23-s − 6/5·25-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 + 0.894·45-s − 0.291·47-s − 0.288·48-s + 1/7·49-s − 0.274·53-s + 0.269·55-s + 0.516·60-s − 1/8·64-s − 0.481·69-s + 1.18·71-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: yes
Self-dual: yes
Analytic rank: \(1\)
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$C_2$ \( 1 + T^{2} \)
3$C_1$ \( ( 1 + T )^{2} \)
7$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
11$C_1$ \( 1 - T \)
good5$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + p T^{2} ) \)
13$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \)
17$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \)
19$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
23$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
29$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \)
31$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
37$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
41$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \)
43$C_2^2$ \( 1 + 46 T^{2} + p^{2} T^{4} \)
47$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 2 T + p T^{2} ) \)
53$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
59$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
61$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \)
67$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
71$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
73$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \)
79$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \)
83$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
89$C_2$$\times$$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 - 10 T + p T^{2} ) \)
97$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
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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

−7.41770998509546734037712728327, −6.81539110196714991218053395773, −6.66141965171668532695097678701, −6.00887630839035476264421082560, −5.79295625950230208692917141974, −5.41632630190641561348112455249, −4.99232032691374663214855023871, −4.55759299848791033942009421313, −3.91165220490796163542037045395, −3.69912265401621314150175657358, −2.88447614349323662960812654781, −2.11455881782595400347484153694, −1.67832160592333745510581620712, −0.927973759361932525745661905814, 0, 0.927973759361932525745661905814, 1.67832160592333745510581620712, 2.11455881782595400347484153694, 2.88447614349323662960812654781, 3.69912265401621314150175657358, 3.91165220490796163542037045395, 4.55759299848791033942009421313, 4.99232032691374663214855023871, 5.41632630190641561348112455249, 5.79295625950230208692917141974, 6.00887630839035476264421082560, 6.66141965171668532695097678701, 6.81539110196714991218053395773, 7.41770998509546734037712728327

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