Properties

Label 4-790272-1.1-c1e2-0-26
Degree $4$
Conductor $790272$
Sign $-1$
Analytic cond. $50.3884$
Root an. cond. $2.66429$
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 − 2·5-s − 7-s + 9-s + 4·15-s + 2·17-s + 2·21-s − 6·25-s + 4·27-s + 2·35-s − 8·37-s + 10·41-s − 4·43-s − 2·45-s + 8·47-s + 49-s − 4·51-s − 4·59-s − 63-s + 4·67-s + 12·75-s − 4·79-s − 11·81-s + 20·83-s − 4·85-s − 22·89-s − 2·101-s + ⋯
L(s)  = 1  − 1.15·3-s − 0.894·5-s − 0.377·7-s + 1/3·9-s + 1.03·15-s + 0.485·17-s + 0.436·21-s − 6/5·25-s + 0.769·27-s + 0.338·35-s − 1.31·37-s + 1.56·41-s − 0.609·43-s − 0.298·45-s + 1.16·47-s + 1/7·49-s − 0.560·51-s − 0.520·59-s − 0.125·63-s + 0.488·67-s + 1.38·75-s − 0.450·79-s − 1.22·81-s + 2.19·83-s − 0.433·85-s − 2.33·89-s − 0.199·101-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 790272 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 790272 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(790272\)    =    \(2^{8} \cdot 3^{2} \cdot 7^{3}\)
Sign: $-1$
Analytic conductor: \(50.3884\)
Root analytic conductor: \(2.66429\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 790272,\ (\ :1/2, 1/2),\ -1)\)

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 \( 1 \)
3$C_2$ \( 1 + 2 T + p T^{2} \)
7$C_1$ \( 1 + T \)
good5$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 2 T + p T^{2} ) \)
11$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
13$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \)
17$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + p T^{2} ) \)
19$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
23$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
29$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \)
31$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
37$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
41$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 - 2 T + p T^{2} ) \)
43$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \)
47$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} ) \)
53$C_2^2$ \( 1 - 74 T^{2} + p^{2} T^{4} \)
59$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \)
61$C_2^2$ \( 1 - 42 T^{2} + p^{2} T^{4} \)
67$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
71$C_2^2$ \( 1 - 110 T^{2} + p^{2} T^{4} \)
73$C_2^2$ \( 1 - 74 T^{2} + p^{2} T^{4} \)
79$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
83$C_2$$\times$$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 - 6 T + p T^{2} ) \)
89$C_2$$\times$$C_2$ \( ( 1 + 10 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
97$C_2^2$ \( 1 + 30 T^{2} + p^{2} T^{4} \)
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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

−8.006810999334306246014221316282, −7.41679787328512967221220736949, −7.26833237112847662899203072491, −6.57889005868593029480881427029, −6.25217444268487149618739817919, −5.66400557363162770922137856429, −5.47422062256893143755282411218, −4.84514450356068525065576764403, −4.27252669055871372625228478133, −3.86681736268164082609446233755, −3.32746840802018450087510866921, −2.67603562035073970243533236266, −1.83169799468905694838870970035, −0.833309481322196499491609488437, 0, 0.833309481322196499491609488437, 1.83169799468905694838870970035, 2.67603562035073970243533236266, 3.32746840802018450087510866921, 3.86681736268164082609446233755, 4.27252669055871372625228478133, 4.84514450356068525065576764403, 5.47422062256893143755282411218, 5.66400557363162770922137856429, 6.25217444268487149618739817919, 6.57889005868593029480881427029, 7.26833237112847662899203072491, 7.41679787328512967221220736949, 8.006810999334306246014221316282

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