Properties

Label 2-4022-1.1-c1-0-78
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 + 0.761·3-s + 4-s + 0.666·5-s + 0.761·6-s − 1.04·7-s + 8-s − 2.41·9-s + 0.666·10-s + 5.59·11-s + 0.761·12-s + 5.65·13-s − 1.04·14-s + 0.507·15-s + 16-s + 2.95·17-s − 2.41·18-s − 7.03·19-s + 0.666·20-s − 0.797·21-s + 5.59·22-s + 7.83·23-s + 0.761·24-s − 4.55·25-s + 5.65·26-s − 4.12·27-s − 1.04·28-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.439·3-s + 0.5·4-s + 0.297·5-s + 0.310·6-s − 0.395·7-s + 0.353·8-s − 0.806·9-s + 0.210·10-s + 1.68·11-s + 0.219·12-s + 1.56·13-s − 0.279·14-s + 0.131·15-s + 0.250·16-s + 0.717·17-s − 0.570·18-s − 1.61·19-s + 0.148·20-s − 0.174·21-s + 1.19·22-s + 1.63·23-s + 0.155·24-s − 0.911·25-s + 1.10·26-s − 0.794·27-s − 0.197·28-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

\(L(1)\) \(\approx\) \(4.101924923\)
\(L(\frac12)\) \(\approx\) \(4.101924923\)
\(L(\frac{3}{2})\) 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 \( 1 - 0.761T + 3T^{2} \)
5 \( 1 - 0.666T + 5T^{2} \)
7 \( 1 + 1.04T + 7T^{2} \)
11 \( 1 - 5.59T + 11T^{2} \)
13 \( 1 - 5.65T + 13T^{2} \)
17 \( 1 - 2.95T + 17T^{2} \)
19 \( 1 + 7.03T + 19T^{2} \)
23 \( 1 - 7.83T + 23T^{2} \)
29 \( 1 + 6.27T + 29T^{2} \)
31 \( 1 - 5.53T + 31T^{2} \)
37 \( 1 - 10.8T + 37T^{2} \)
41 \( 1 - 2.70T + 41T^{2} \)
43 \( 1 + 9.38T + 43T^{2} \)
47 \( 1 - 4.88T + 47T^{2} \)
53 \( 1 - 3.98T + 53T^{2} \)
59 \( 1 - 1.28T + 59T^{2} \)
61 \( 1 + 2.84T + 61T^{2} \)
67 \( 1 + 14.0T + 67T^{2} \)
71 \( 1 - 14.7T + 71T^{2} \)
73 \( 1 + 3.54T + 73T^{2} \)
79 \( 1 - 8.00T + 79T^{2} \)
83 \( 1 + 0.521T + 83T^{2} \)
89 \( 1 - 10.7T + 89T^{2} \)
97 \( 1 - 2.41T + 97T^{2} \)
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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

−8.541489069543443444746703394075, −7.73739808625497890112768747526, −6.60549791432461589967169961302, −6.24233654850864052531906869132, −5.68273663699535104234145156563, −4.47699436992936741546912279379, −3.72573794956768045653528805671, −3.21191934756296576628803361709, −2.11752118017256374851224261768, −1.09743226725543604043978902483, 1.09743226725543604043978902483, 2.11752118017256374851224261768, 3.21191934756296576628803361709, 3.72573794956768045653528805671, 4.47699436992936741546912279379, 5.68273663699535104234145156563, 6.24233654850864052531906869132, 6.60549791432461589967169961302, 7.73739808625497890112768747526, 8.541489069543443444746703394075

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