Properties

Label 2-3311-3311.3310-c0-0-25
Degree $2$
Conductor $3311$
Sign $1$
Analytic cond. $1.65240$
Root an. cond. $1.28545$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.53·2-s − 1.96·3-s + 1.34·4-s + 1.28·5-s − 3.01·6-s + 7-s + 0.532·8-s + 2.87·9-s + 1.96·10-s − 11-s − 2.65·12-s + 1.73·13-s + 1.53·14-s − 2.53·15-s − 0.532·16-s − 0.684·17-s + 4.41·18-s + 1.73·20-s − 1.96·21-s − 1.53·22-s + 23-s − 1.04·24-s + 0.652·25-s + 2.65·26-s − 3.70·27-s + 1.34·28-s − 0.347·29-s + ⋯
L(s)  = 1  + 1.53·2-s − 1.96·3-s + 1.34·4-s + 1.28·5-s − 3.01·6-s + 7-s + 0.532·8-s + 2.87·9-s + 1.96·10-s − 11-s − 2.65·12-s + 1.73·13-s + 1.53·14-s − 2.53·15-s − 0.532·16-s − 0.684·17-s + 4.41·18-s + 1.73·20-s − 1.96·21-s − 1.53·22-s + 23-s − 1.04·24-s + 0.652·25-s + 2.65·26-s − 3.70·27-s + 1.34·28-s − 0.347·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3311 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3311 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3311\)    =    \(7 \cdot 11 \cdot 43\)
Sign: $1$
Analytic conductor: \(1.65240\)
Root analytic conductor: \(1.28545\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3311} (3310, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 3311,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.209020093\)
\(L(\frac12)\) \(\approx\) \(2.209020093\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 - T \)
11 \( 1 + T \)
43 \( 1 + T \)
good2 \( 1 - 1.53T + T^{2} \)
3 \( 1 + 1.96T + T^{2} \)
5 \( 1 - 1.28T + T^{2} \)
13 \( 1 - 1.73T + T^{2} \)
17 \( 1 + 0.684T + T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 - T + T^{2} \)
29 \( 1 + 0.347T + T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 - 0.684T + T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + 1.87T + T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 - 1.53T + T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 - 1.96T + T^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 - T^{2} \)
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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

−8.898928529603094074131830040419, −7.68804865044099508765296159335, −6.63692396953606996417809767637, −6.25759219855834796251004827812, −5.60712571370245991408071246588, −5.11295444155801005289922682034, −4.65302670366272961243435162874, −3.66365814683845920905785382330, −2.19715821318105808118061550971, −1.31928300792488560915036110146, 1.31928300792488560915036110146, 2.19715821318105808118061550971, 3.66365814683845920905785382330, 4.65302670366272961243435162874, 5.11295444155801005289922682034, 5.60712571370245991408071246588, 6.25759219855834796251004827812, 6.63692396953606996417809767637, 7.68804865044099508765296159335, 8.898928529603094074131830040419

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