Properties

Label 4-7889-1.1-c1e2-0-0
Degree $4$
Conductor $7889$
Sign $1$
Analytic cond. $0.503009$
Root an. cond. $0.842158$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·4-s + 7-s + 4·9-s + 3·11-s + 12·16-s + 2·23-s − 25-s − 4·28-s − 3·29-s − 16·36-s − 5·37-s + 7·43-s − 12·44-s + 49-s − 3·53-s + 4·63-s − 32·64-s − 17·67-s + 3·77-s − 2·79-s + 7·81-s − 8·92-s + 12·99-s + 4·100-s − 15·107-s + 4·109-s + 12·112-s + ⋯
L(s)  = 1  − 2·4-s + 0.377·7-s + 4/3·9-s + 0.904·11-s + 3·16-s + 0.417·23-s − 1/5·25-s − 0.755·28-s − 0.557·29-s − 8/3·36-s − 0.821·37-s + 1.06·43-s − 1.80·44-s + 1/7·49-s − 0.412·53-s + 0.503·63-s − 4·64-s − 2.07·67-s + 0.341·77-s − 0.225·79-s + 7/9·81-s − 0.834·92-s + 1.20·99-s + 2/5·100-s − 1.45·107-s + 0.383·109-s + 1.13·112-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7889 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7889 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(7889\)    =    \(7^{3} \cdot 23\)
Sign: $1$
Analytic conductor: \(0.503009\)
Root analytic conductor: \(0.842158\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{7889} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 7889,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7285964346\)
\(L(\frac12)\) \(\approx\) \(0.7285964346\)
\(L(\frac{3}{2})\) 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)$
bad7$C_1$ \( 1 - T \)
23$C_1$$\times$$C_2$ \( ( 1 + T )( 1 - 3 T + p T^{2} ) \)
good2$C_2$ \( ( 1 + p T^{2} )^{2} \)
3$C_2^2$ \( 1 - 4 T^{2} + p^{2} T^{4} \)
5$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
11$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
13$C_2^2$ \( 1 - 8 T^{2} + p^{2} T^{4} \)
17$C_2^2$ \( 1 - 29 T^{2} + p^{2} T^{4} \)
19$C_2^2$ \( 1 + 7 T^{2} + p^{2} T^{4} \)
29$C_2$$\times$$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
31$C_2^2$ \( 1 + 28 T^{2} + p^{2} T^{4} \)
37$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
41$C_2^2$ \( 1 - 8 T^{2} + p^{2} T^{4} \)
43$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + T + p T^{2} ) \)
47$C_2^2$ \( 1 - 68 T^{2} + p^{2} T^{4} \)
53$C_2$$\times$$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
59$C_2^2$ \( 1 - 44 T^{2} + p^{2} T^{4} \)
61$C_2^2$ \( 1 - 5 T^{2} + p^{2} T^{4} \)
67$C_2$$\times$$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 13 T + p T^{2} ) \)
71$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \)
73$C_2^2$ \( 1 - 20 T^{2} + p^{2} T^{4} \)
79$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
83$C_2^2$ \( 1 + 49 T^{2} + p^{2} T^{4} \)
89$C_2^2$ \( 1 + 115 T^{2} + p^{2} T^{4} \)
97$C_2^2$ \( 1 + 13 T^{2} + p^{2} T^{4} \)
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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

−11.99293803187692863612848935232, −11.14370262460570851054722692030, −10.37985969179056663615225401949, −10.01043108557317592181541362170, −9.302950362740022149544439312301, −9.068524916739988914708835843847, −8.431996768551471719612410029961, −7.67853877069994036870786216970, −7.18497874974977484189626933572, −6.15817669467200672583958207324, −5.37520122098436180547431489404, −4.60992452287576728117933079355, −4.19362036447808193879643224019, −3.47606940522207266836370919136, −1.37377845136551597379361654847, 1.37377845136551597379361654847, 3.47606940522207266836370919136, 4.19362036447808193879643224019, 4.60992452287576728117933079355, 5.37520122098436180547431489404, 6.15817669467200672583958207324, 7.18497874974977484189626933572, 7.67853877069994036870786216970, 8.431996768551471719612410029961, 9.068524916739988914708835843847, 9.302950362740022149544439312301, 10.01043108557317592181541362170, 10.37985969179056663615225401949, 11.14370262460570851054722692030, 11.99293803187692863612848935232

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