Properties

Label 2-3-1.1-c11-0-0
Degree $2$
Conductor $3$
Sign $1$
Analytic cond. $2.30502$
Root an. cond. $1.51823$
Motivic weight $11$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 78·2-s − 243·3-s + 4.03e3·4-s − 5.37e3·5-s − 1.89e4·6-s − 2.77e4·7-s + 1.55e5·8-s + 5.90e4·9-s − 4.18e5·10-s + 6.37e5·11-s − 9.80e5·12-s + 7.66e5·13-s − 2.16e6·14-s + 1.30e6·15-s + 3.82e6·16-s + 3.08e6·17-s + 4.60e6·18-s − 1.95e7·19-s − 2.16e7·20-s + 6.74e6·21-s + 4.97e7·22-s + 1.53e7·23-s − 3.76e7·24-s − 1.99e7·25-s + 5.97e7·26-s − 1.43e7·27-s − 1.12e8·28-s + ⋯
L(s)  = 1  + 1.72·2-s − 0.577·3-s + 1.97·4-s − 0.768·5-s − 0.995·6-s − 0.624·7-s + 1.67·8-s + 1/3·9-s − 1.32·10-s + 1.19·11-s − 1.13·12-s + 0.572·13-s − 1.07·14-s + 0.443·15-s + 0.912·16-s + 0.526·17-s + 0.574·18-s − 1.80·19-s − 1.51·20-s + 0.360·21-s + 2.05·22-s + 0.496·23-s − 0.965·24-s − 0.409·25-s + 0.986·26-s − 0.192·27-s − 1.23·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3\)
Sign: $1$
Analytic conductor: \(2.30502\)
Root analytic conductor: \(1.51823\)
Motivic weight: \(11\)
Rational: yes
Arithmetic: yes
Character: $\chi_{3} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 3,\ (\ :11/2),\ 1)\)

Particular Values

\(L(6)\) \(\approx\) \(2.317708444\)
\(L(\frac12)\) \(\approx\) \(2.317708444\)
\(L(\frac{13}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + p^{5} T \)
good2 \( 1 - 39 p T + p^{11} T^{2} \)
5 \( 1 + 1074 p T + p^{11} T^{2} \)
7 \( 1 + 27760 T + p^{11} T^{2} \)
11 \( 1 - 637836 T + p^{11} T^{2} \)
13 \( 1 - 766214 T + p^{11} T^{2} \)
17 \( 1 - 3084354 T + p^{11} T^{2} \)
19 \( 1 + 1026916 p T + p^{11} T^{2} \)
23 \( 1 - 15312360 T + p^{11} T^{2} \)
29 \( 1 - 10751262 T + p^{11} T^{2} \)
31 \( 1 + 50937400 T + p^{11} T^{2} \)
37 \( 1 - 664740830 T + p^{11} T^{2} \)
41 \( 1 - 898833450 T + p^{11} T^{2} \)
43 \( 1 + 957947188 T + p^{11} T^{2} \)
47 \( 1 + 1555741344 T + p^{11} T^{2} \)
53 \( 1 - 3792417030 T + p^{11} T^{2} \)
59 \( 1 - 555306924 T + p^{11} T^{2} \)
61 \( 1 - 4950420998 T + p^{11} T^{2} \)
67 \( 1 - 5292399284 T + p^{11} T^{2} \)
71 \( 1 + 14831086248 T + p^{11} T^{2} \)
73 \( 1 - 13971005210 T + p^{11} T^{2} \)
79 \( 1 - 3720542360 T + p^{11} T^{2} \)
83 \( 1 - 8768454036 T + p^{11} T^{2} \)
89 \( 1 + 25472769174 T + p^{11} T^{2} \)
97 \( 1 + 39092494846 T + p^{11} T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−23.58034982785144348964337370029, −22.81545721493593948533526458611, −21.44601855398217842094957197745, −19.55060201813449012653246471853, −16.44536996040283797615892064964, −14.88731530558822093238273533402, −12.84784253531214435090531656458, −11.44100936879895756795295476050, −6.38978670338428742867669092804, −3.98650105120338668052543062604, 3.98650105120338668052543062604, 6.38978670338428742867669092804, 11.44100936879895756795295476050, 12.84784253531214435090531656458, 14.88731530558822093238273533402, 16.44536996040283797615892064964, 19.55060201813449012653246471853, 21.44601855398217842094957197745, 22.81545721493593948533526458611, 23.58034982785144348964337370029

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