Properties

Label 2-4022-1.1-c1-0-45
Degree $2$
Conductor $4022$
Sign $1$
Analytic cond. $32.1158$
Root an. cond. $5.66708$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 1.87·3-s + 4-s − 4.12·5-s − 1.87·6-s + 2.22·7-s − 8-s + 0.529·9-s + 4.12·10-s + 3.54·11-s + 1.87·12-s + 2.44·13-s − 2.22·14-s − 7.74·15-s + 16-s + 2.66·17-s − 0.529·18-s + 3.11·19-s − 4.12·20-s + 4.17·21-s − 3.54·22-s − 0.627·23-s − 1.87·24-s + 12.0·25-s − 2.44·26-s − 4.64·27-s + 2.22·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 1.08·3-s + 0.5·4-s − 1.84·5-s − 0.766·6-s + 0.840·7-s − 0.353·8-s + 0.176·9-s + 1.30·10-s + 1.06·11-s + 0.542·12-s + 0.678·13-s − 0.594·14-s − 2.00·15-s + 0.250·16-s + 0.645·17-s − 0.124·18-s + 0.715·19-s − 0.922·20-s + 0.911·21-s − 0.755·22-s − 0.130·23-s − 0.383·24-s + 2.40·25-s − 0.479·26-s − 0.893·27-s + 0.420·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4022 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4022 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4022\)    =    \(2 \cdot 2011\)
Sign: $1$
Analytic conductor: \(32.1158\)
Root analytic conductor: \(5.66708\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 4022,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.698473621\)
\(L(\frac12)\) \(\approx\) \(1.698473621\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
2011 \( 1 - T \)
good3 \( 1 - 1.87T + 3T^{2} \)
5 \( 1 + 4.12T + 5T^{2} \)
7 \( 1 - 2.22T + 7T^{2} \)
11 \( 1 - 3.54T + 11T^{2} \)
13 \( 1 - 2.44T + 13T^{2} \)
17 \( 1 - 2.66T + 17T^{2} \)
19 \( 1 - 3.11T + 19T^{2} \)
23 \( 1 + 0.627T + 23T^{2} \)
29 \( 1 - 2.51T + 29T^{2} \)
31 \( 1 - 1.49T + 31T^{2} \)
37 \( 1 + 1.70T + 37T^{2} \)
41 \( 1 + 9.45T + 41T^{2} \)
43 \( 1 + 12.0T + 43T^{2} \)
47 \( 1 + 0.282T + 47T^{2} \)
53 \( 1 - 11.3T + 53T^{2} \)
59 \( 1 - 0.624T + 59T^{2} \)
61 \( 1 - 12.3T + 61T^{2} \)
67 \( 1 - 6.47T + 67T^{2} \)
71 \( 1 + 1.78T + 71T^{2} \)
73 \( 1 - 3.55T + 73T^{2} \)
79 \( 1 + 6.87T + 79T^{2} \)
83 \( 1 + 6.71T + 83T^{2} \)
89 \( 1 - 14.4T + 89T^{2} \)
97 \( 1 - 8.90T + 97T^{2} \)
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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

−8.318901137971840055737388836909, −8.116788504118860431113398582409, −7.23518924391018244750642096455, −6.72343944543760214791203390745, −5.39264540029753718804349493427, −4.38194243233455896004695354458, −3.52992084782132746928138140610, −3.22845628759779908211959571570, −1.82467127400991978848018959486, −0.820456304790009708789165415209, 0.820456304790009708789165415209, 1.82467127400991978848018959486, 3.22845628759779908211959571570, 3.52992084782132746928138140610, 4.38194243233455896004695354458, 5.39264540029753718804349493427, 6.72343944543760214791203390745, 7.23518924391018244750642096455, 8.116788504118860431113398582409, 8.318901137971840055737388836909

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