Properties

Label 4-2900-1.1-c1e2-0-1
Degree $4$
Conductor $2900$
Sign $1$
Analytic cond. $0.184906$
Root an. cond. $0.655749$
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-s − 5-s − 4·9-s + 3·11-s + 16-s − 9·19-s − 20-s − 4·25-s + 10·29-s + 31-s − 4·36-s − 7·41-s + 3·44-s + 4·45-s + 49-s − 3·55-s + 8·59-s − 2·61-s + 64-s + 25·71-s − 9·76-s − 79-s − 80-s + 7·81-s − 9·89-s + 9·95-s − 12·99-s + ⋯
L(s)  = 1  + 1/2·4-s − 0.447·5-s − 4/3·9-s + 0.904·11-s + 1/4·16-s − 2.06·19-s − 0.223·20-s − 4/5·25-s + 1.85·29-s + 0.179·31-s − 2/3·36-s − 1.09·41-s + 0.452·44-s + 0.596·45-s + 1/7·49-s − 0.404·55-s + 1.04·59-s − 0.256·61-s + 1/8·64-s + 2.96·71-s − 1.03·76-s − 0.112·79-s − 0.111·80-s + 7/9·81-s − 0.953·89-s + 0.923·95-s − 1.20·99-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(2900\)    =    \(2^{2} \cdot 5^{2} \cdot 29\)
Sign: $1$
Analytic conductor: \(0.184906\)
Root analytic conductor: \(0.655749\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 2900,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7115995980\)
\(L(\frac12)\) \(\approx\) \(0.7115995980\)
\(L(\frac{3}{2})\) 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$ \( ( 1 - T )( 1 + T ) \)
5$C_2$ \( 1 + T + p T^{2} \)
29$C_1$$\times$$C_2$ \( ( 1 - T )( 1 - 9 T + p T^{2} ) \)
good3$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \)
7$C_2^2$ \( 1 - T^{2} + p^{2} T^{4} \)
11$C_2$$\times$$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
13$C_2^2$ \( 1 - 9 T^{2} + p^{2} T^{4} \)
17$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 + T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
23$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
31$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
37$C_2^2$ \( 1 - 40 T^{2} + p^{2} T^{4} \)
41$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 7 T + p T^{2} ) \)
43$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 94 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
61$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
67$C_2^2$ \( 1 - 73 T^{2} + p^{2} T^{4} \)
71$C_2$$\times$$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 - 9 T + p T^{2} ) \)
73$C_2^2$ \( 1 + 60 T^{2} + p^{2} T^{4} \)
79$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 11 T + p T^{2} ) \)
83$C_2^2$ \( 1 + 131 T^{2} + p^{2} T^{4} \)
89$C_2$$\times$$C_2$ \( ( 1 + T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
97$C_2^2$ \( 1 - 86 T^{2} + p^{2} T^{4} \)
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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

−12.62612934929032554678336338024, −12.22336667625757416270395253552, −11.60381783260442565766016521761, −11.22633663941716090666286842996, −10.55578264536791298876246669639, −9.910364646883300315518125241244, −8.976851365462203903291224173433, −8.375065750562238886292296086356, −8.066384233774645269096311711156, −6.77891256781486307782173841295, −6.48142065792258831297629434451, −5.62780188754507505773141895568, −4.51474613651913205199474225191, −3.57421905812722368910314872805, −2.37516654802028633890714464243, 2.37516654802028633890714464243, 3.57421905812722368910314872805, 4.51474613651913205199474225191, 5.62780188754507505773141895568, 6.48142065792258831297629434451, 6.77891256781486307782173841295, 8.066384233774645269096311711156, 8.375065750562238886292296086356, 8.976851365462203903291224173433, 9.910364646883300315518125241244, 10.55578264536791298876246669639, 11.22633663941716090666286842996, 11.60381783260442565766016521761, 12.22336667625757416270395253552, 12.62612934929032554678336338024

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