Properties

Degree $4$
Conductor $213444$
Sign $-1$
Motivic weight $1$
Primitive no
Self-dual yes
Analytic rank $1$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s − 2·3-s + 3·4-s + 4·6-s − 4·8-s + 9-s − 6·12-s + 5·16-s + 12·17-s − 2·18-s + 8·24-s − 10·25-s + 4·27-s − 12·29-s − 8·31-s − 6·32-s − 24·34-s + 3·36-s + 4·37-s + 12·41-s − 10·48-s + 49-s + 20·50-s − 24·51-s − 8·54-s + 24·58-s + 16·62-s + ⋯
L(s)  = 1  − 1.41·2-s − 1.15·3-s + 3/2·4-s + 1.63·6-s − 1.41·8-s + 1/3·9-s − 1.73·12-s + 5/4·16-s + 2.91·17-s − 0.471·18-s + 1.63·24-s − 2·25-s + 0.769·27-s − 2.22·29-s − 1.43·31-s − 1.06·32-s − 4.11·34-s + 1/2·36-s + 0.657·37-s + 1.87·41-s − 1.44·48-s + 1/7·49-s + 2.82·50-s − 3.36·51-s − 1.08·54-s + 3.15·58-s + 2.03·62-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 213444 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 213444 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(213444\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2} \cdot 11^{2}\)
Sign: $-1$
Motivic weight: \(1\)
Character: $\chi_{213444} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: no
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 213444,\ (\ :1/2, 1/2),\ -1)\)

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$C_1$ \( ( 1 + T )^{2} \)
3$C_2$ \( 1 + 2 T + p T^{2} \)
7$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
11$C_2$ \( 1 + p T^{2} \)
good5$C_2$ \( ( 1 + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
17$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
23$C_2$ \( ( 1 + p T^{2} )^{2} \)
29$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
47$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
59$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
61$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
67$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
83$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
89$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
97$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
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   \(L(s) = \displaystyle\prod_{\mathfrak{p}\ \mathrm{bad}} (1- a(\mathfrak{p}) (N\mathfrak{p})^{-s})^{-1} \prod_{\mathfrak{p}\ \mathrm{good}} (1- a(\mathfrak{p}) (N\mathfrak{p})^{-s} + (N\mathfrak{p})^{-2s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.854560326398960886018318456293, −8.341492939347990953072051690786, −7.57571100088867902110310233811, −7.55188433888647441135901584446, −7.30018879519451360703630309495, −6.19937662517262739965360608374, −5.94883440657990935524760577420, −5.57928681742950427486583645839, −5.25092731809634987993751351760, −4.06855698427217288533389536167, −3.60551513627682819583067906026, −2.81618810622184647230747429130, −1.84109947013474715854005945923, −1.11665210476803236518648751929, 0, 1.11665210476803236518648751929, 1.84109947013474715854005945923, 2.81618810622184647230747429130, 3.60551513627682819583067906026, 4.06855698427217288533389536167, 5.25092731809634987993751351760, 5.57928681742950427486583645839, 5.94883440657990935524760577420, 6.19937662517262739965360608374, 7.30018879519451360703630309495, 7.55188433888647441135901584446, 7.57571100088867902110310233811, 8.341492939347990953072051690786, 8.854560326398960886018318456293

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