Properties

Label 4-7497-1.1-c1e2-0-2
Degree $4$
Conductor $7497$
Sign $-1$
Analytic cond. $0.478015$
Root an. cond. $0.831496$
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  − 3-s − 4-s − 3·5-s − 7-s − 2·9-s + 12-s + 3·15-s − 3·16-s − 5·17-s + 3·20-s + 21-s − 25-s + 5·27-s + 28-s + 3·35-s + 2·36-s + 7·37-s + 9·41-s − 5·43-s + 6·45-s + 3·47-s + 3·48-s − 6·49-s + 5·51-s − 15·59-s − 3·60-s + 2·63-s + ⋯
L(s)  = 1  − 0.577·3-s − 1/2·4-s − 1.34·5-s − 0.377·7-s − 2/3·9-s + 0.288·12-s + 0.774·15-s − 3/4·16-s − 1.21·17-s + 0.670·20-s + 0.218·21-s − 1/5·25-s + 0.962·27-s + 0.188·28-s + 0.507·35-s + 1/3·36-s + 1.15·37-s + 1.40·41-s − 0.762·43-s + 0.894·45-s + 0.437·47-s + 0.433·48-s − 6/7·49-s + 0.700·51-s − 1.95·59-s − 0.387·60-s + 0.251·63-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7497 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7497 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(7497\)    =    \(3^{2} \cdot 7^{2} \cdot 17\)
Sign: $-1$
Analytic conductor: \(0.478015\)
Root analytic conductor: \(0.831496\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{7497} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 7497,\ (\ :1/2, 1/2),\ -1)\)

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)$
bad3$C_2$ \( 1 + T + p T^{2} \)
7$C_2$ \( 1 + T + p T^{2} \)
17$C_1$$\times$$C_2$ \( ( 1 - T )( 1 + 6 T + p T^{2} ) \)
good2$C_2^2$ \( 1 + T^{2} + p^{2} T^{4} \)
5$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 3 T + p T^{2} ) \)
11$C_2^2$ \( 1 - 5 T^{2} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
19$C_2^2$ \( 1 - 20 T^{2} + p^{2} T^{4} \)
23$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
31$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
37$C_2$$\times$$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 - 2 T + p T^{2} ) \)
41$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
43$C_2$$\times$$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
47$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
53$C_2^2$ \( 1 + 73 T^{2} + p^{2} T^{4} \)
59$C_2$$\times$$C_2$ \( ( 1 + 3 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
61$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \)
67$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
71$C_2^2$ \( 1 + 13 T^{2} + p^{2} T^{4} \)
73$C_2^2$ \( 1 - 20 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
83$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 9 T + p T^{2} ) \)
89$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 9 T + p T^{2} ) \)
97$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \)
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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

−11.34508854331484642376991905245, −11.25707327342411306597413346788, −10.63840718641635641748219944339, −9.724497989197171679936034648949, −9.159373794025726824108071661899, −8.658397679023640077898912500737, −7.947949238226173733402460614730, −7.47623684101418828278444982747, −6.52661885747015299786663397245, −6.13190900282516682859975203745, −5.07572336752134301502355514955, −4.42062313631998468504582230325, −3.79648311689978530464002654712, −2.65092696918366253732453911613, 0, 2.65092696918366253732453911613, 3.79648311689978530464002654712, 4.42062313631998468504582230325, 5.07572336752134301502355514955, 6.13190900282516682859975203745, 6.52661885747015299786663397245, 7.47623684101418828278444982747, 7.947949238226173733402460614730, 8.658397679023640077898912500737, 9.159373794025726824108071661899, 9.724497989197171679936034648949, 10.63840718641635641748219944339, 11.25707327342411306597413346788, 11.34508854331484642376991905245

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