Properties

Label 2-3311-3311.3310-c0-0-23
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  − 1.87·2-s − 1.53·3-s + 2.53·4-s + 1.87·5-s + 2.87·6-s + 7-s − 2.87·8-s + 1.34·9-s − 3.53·10-s + 11-s − 3.87·12-s + 13-s − 1.87·14-s − 2.87·15-s + 2.87·16-s − 0.347·17-s − 2.53·18-s + 4.75·20-s − 1.53·21-s − 1.87·22-s − 23-s + 4.41·24-s + 2.53·25-s − 1.87·26-s − 0.532·27-s + 2.53·28-s + 1.53·29-s + ⋯
L(s)  = 1  − 1.87·2-s − 1.53·3-s + 2.53·4-s + 1.87·5-s + 2.87·6-s + 7-s − 2.87·8-s + 1.34·9-s − 3.53·10-s + 11-s − 3.87·12-s + 13-s − 1.87·14-s − 2.87·15-s + 2.87·16-s − 0.347·17-s − 2.53·18-s + 4.75·20-s − 1.53·21-s − 1.87·22-s − 23-s + 4.41·24-s + 2.53·25-s − 1.87·26-s − 0.532·27-s + 2.53·28-s + 1.53·29-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

\(L(\frac{1}{2})\) \(\approx\) \(0.6118737738\)
\(L(\frac12)\) \(\approx\) \(0.6118737738\)
\(L(1)\) 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 \( 1 + 1.87T + T^{2} \)
3 \( 1 + 1.53T + T^{2} \)
5 \( 1 - 1.87T + T^{2} \)
13 \( 1 - T + T^{2} \)
17 \( 1 + 0.347T + T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + T + T^{2} \)
29 \( 1 - 1.53T + T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + 0.347T + T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 - 0.347T + T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 + 1.87T + T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + 1.53T + T^{2} \)
89 \( 1 + 2T + T^{2} \)
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

−8.820907628785638015877775555640, −8.425645043108580951612449419865, −7.19426177888150238745669960175, −6.52048485066540500825120694662, −6.06302169485209141663847961218, −5.55814748472375137751210012237, −4.42399214841441578022954060346, −2.56070836490988475395305869405, −1.50771114210261946106002010694, −1.18542852239258822122214858277, 1.18542852239258822122214858277, 1.50771114210261946106002010694, 2.56070836490988475395305869405, 4.42399214841441578022954060346, 5.55814748472375137751210012237, 6.06302169485209141663847961218, 6.52048485066540500825120694662, 7.19426177888150238745669960175, 8.425645043108580951612449419865, 8.820907628785638015877775555640

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