Properties

Label 4-260876-1.1-c1e2-0-8
Degree $4$
Conductor $260876$
Sign $-1$
Analytic cond. $16.6336$
Root an. cond. $2.01951$
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  + 4-s − 2·9-s − 11-s + 16-s − 16·23-s + 10·25-s − 2·36-s + 4·37-s − 44-s − 7·49-s − 28·53-s + 64-s − 8·67-s − 5·81-s − 16·92-s + 2·99-s + 10·100-s + 4·113-s + 121-s + 127-s + 131-s + 137-s + 139-s − 2·144-s + 4·148-s + 149-s + 151-s + ⋯
L(s)  = 1  + 1/2·4-s − 2/3·9-s − 0.301·11-s + 1/4·16-s − 3.33·23-s + 2·25-s − 1/3·36-s + 0.657·37-s − 0.150·44-s − 49-s − 3.84·53-s + 1/8·64-s − 0.977·67-s − 5/9·81-s − 1.66·92-s + 0.201·99-s + 100-s + 0.376·113-s + 1/11·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 1/6·144-s + 0.328·148-s + 0.0819·149-s + 0.0813·151-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 260876 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 260876 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(260876\)    =    \(2^{2} \cdot 7^{2} \cdot 11^{3}\)
Sign: $-1$
Analytic conductor: \(16.6336\)
Root analytic conductor: \(2.01951\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{260876} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 260876,\ (\ :1/2, 1/2),\ -1)\)

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$ \( ( 1 - T )( 1 + T ) \)
7$C_2$ \( 1 + p T^{2} \)
11$C_1$ \( 1 + T \)
good3$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
5$C_2$ \( ( 1 - p T^{2} )^{2} \)
13$C_2^2$ \( 1 + 18 T^{2} + p^{2} T^{4} \)
17$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \)
19$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \)
23$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
31$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \)
37$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 + 74 T^{2} + p^{2} T^{4} \)
43$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
47$C_2^2$ \( 1 - 86 T^{2} + p^{2} T^{4} \)
53$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \)
59$C_2^2$ \( 1 - 46 T^{2} + p^{2} T^{4} \)
61$C_2^2$ \( 1 + 50 T^{2} + p^{2} T^{4} \)
67$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 54 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
83$C_2^2$ \( 1 - 122 T^{2} + p^{2} T^{4} \)
89$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \)
97$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
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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

−8.536757738003163472773938294826, −8.166730852936611115960554871947, −7.82021979681689371707293119014, −7.40789779682054330247393673786, −6.56345269675560905791803261211, −6.33341735807205803870658227629, −5.92221856819849171166113388153, −5.32646135494688323743441817683, −4.66180794577029847515972105997, −4.25135230852627933522009234979, −3.35485886243871762322388210983, −2.97182172523047703774195232195, −2.21072101718741916868872670441, −1.50989331984010194201557058291, 0, 1.50989331984010194201557058291, 2.21072101718741916868872670441, 2.97182172523047703774195232195, 3.35485886243871762322388210983, 4.25135230852627933522009234979, 4.66180794577029847515972105997, 5.32646135494688323743441817683, 5.92221856819849171166113388153, 6.33341735807205803870658227629, 6.56345269675560905791803261211, 7.40789779682054330247393673786, 7.82021979681689371707293119014, 8.166730852936611115960554871947, 8.536757738003163472773938294826

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