Properties

Degree $4$
Conductor $7529$
Sign $1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $2$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s − 4·3-s + 4-s − 2·5-s + 8·6-s − 4·7-s + 8·9-s + 4·10-s − 4·11-s − 4·12-s − 3·13-s + 8·14-s + 8·15-s + 16-s − 4·17-s − 16·18-s − 8·19-s − 2·20-s + 16·21-s + 8·22-s + 2·23-s − 2·25-s + 6·26-s − 12·27-s − 4·28-s + 7·29-s − 16·30-s + ⋯
L(s)  = 1  − 1.41·2-s − 2.30·3-s + 1/2·4-s − 0.894·5-s + 3.26·6-s − 1.51·7-s + 8/3·9-s + 1.26·10-s − 1.20·11-s − 1.15·12-s − 0.832·13-s + 2.13·14-s + 2.06·15-s + 1/4·16-s − 0.970·17-s − 3.77·18-s − 1.83·19-s − 0.447·20-s + 3.49·21-s + 1.70·22-s + 0.417·23-s − 2/5·25-s + 1.17·26-s − 2.30·27-s − 0.755·28-s + 1.29·29-s − 2.92·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7529 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7529 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(7529\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{7529} (1, \cdot )$
Sato-Tate group: $\mathrm{USp}(4)$
Primitive: yes
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 7529,\ (\ :1/2, 1/2),\ 1)\)

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)$
bad7529$C_1$$\times$$C_2$ \( ( 1 + T )( 1 + 15 T + p T^{2} ) \)
good2$D_{4}$ \( 1 + p T + 3 T^{2} + p^{2} T^{3} + p^{2} T^{4} \)
3$C_2^2$ \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
5$D_{4}$ \( 1 + 2 T + 6 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
7$D_{4}$ \( 1 + 4 T + 13 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
11$D_{4}$ \( 1 + 4 T + 23 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
13$D_{4}$ \( 1 + 3 T + 4 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 + 4 T + 25 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
19$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \)
23$D_{4}$ \( 1 - 2 T - 5 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 - 7 T + 46 T^{2} - 7 p T^{3} + p^{2} T^{4} \)
31$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
37$D_{4}$ \( 1 - T - 12 T^{2} - p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 - 3 T + 38 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 + 8 T + 42 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 + 8 T + 79 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 + 5 T + 58 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 + 8 T + 87 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 + 9 T + 74 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 + 8 T + 61 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 - 8 T + 81 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 + 2 T + 82 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 + 12 T + 108 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 - 8 T + 37 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 + 6 T + 58 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 + 5 T + 132 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
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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

−17.3851683655, −17.1306588959, −16.6808902232, −16.1776559664, −15.7220148303, −15.4197763233, −14.8863302860, −13.5940036959, −13.1133805410, −12.6543102656, −12.0425430785, −11.8584260472, −10.8766083045, −10.8407796843, −10.0259498006, −9.84676762937, −9.04683032502, −8.23505466260, −7.83518651371, −6.72010859292, −6.51336052150, −5.92367449869, −4.85220090069, −4.43324581354, −2.88547592019, 0, 0, 2.88547592019, 4.43324581354, 4.85220090069, 5.92367449869, 6.51336052150, 6.72010859292, 7.83518651371, 8.23505466260, 9.04683032502, 9.84676762937, 10.0259498006, 10.8407796843, 10.8766083045, 11.8584260472, 12.0425430785, 12.6543102656, 13.1133805410, 13.5940036959, 14.8863302860, 15.4197763233, 15.7220148303, 16.1776559664, 16.6808902232, 17.1306588959, 17.3851683655

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