Properties

Label 2-4011-1.1-c1-0-59
Degree $2$
Conductor $4011$
Sign $1$
Analytic cond. $32.0279$
Root an. cond. $5.65932$
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  − 2.32·2-s + 3-s + 3.39·4-s + 1.73·5-s − 2.32·6-s − 7-s − 3.22·8-s + 9-s − 4.03·10-s + 4.22·11-s + 3.39·12-s + 0.457·13-s + 2.32·14-s + 1.73·15-s + 0.716·16-s − 7.83·17-s − 2.32·18-s + 5.51·19-s + 5.89·20-s − 21-s − 9.80·22-s − 1.27·23-s − 3.22·24-s − 1.97·25-s − 1.06·26-s + 27-s − 3.39·28-s + ⋯
L(s)  = 1  − 1.64·2-s + 0.577·3-s + 1.69·4-s + 0.777·5-s − 0.947·6-s − 0.377·7-s − 1.14·8-s + 0.333·9-s − 1.27·10-s + 1.27·11-s + 0.978·12-s + 0.126·13-s + 0.620·14-s + 0.448·15-s + 0.179·16-s − 1.89·17-s − 0.547·18-s + 1.26·19-s + 1.31·20-s − 0.218·21-s − 2.09·22-s − 0.265·23-s − 0.659·24-s − 0.395·25-s − 0.208·26-s + 0.192·27-s − 0.640·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4011 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4011 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4011\)    =    \(3 \cdot 7 \cdot 191\)
Sign: $1$
Analytic conductor: \(32.0279\)
Root analytic conductor: \(5.65932\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 4011,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.261665103\)
\(L(\frac12)\) \(\approx\) \(1.261665103\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - T \)
7 \( 1 + T \)
191 \( 1 + T \)
good2 \( 1 + 2.32T + 2T^{2} \)
5 \( 1 - 1.73T + 5T^{2} \)
11 \( 1 - 4.22T + 11T^{2} \)
13 \( 1 - 0.457T + 13T^{2} \)
17 \( 1 + 7.83T + 17T^{2} \)
19 \( 1 - 5.51T + 19T^{2} \)
23 \( 1 + 1.27T + 23T^{2} \)
29 \( 1 - 8.13T + 29T^{2} \)
31 \( 1 + 3.47T + 31T^{2} \)
37 \( 1 + 1.49T + 37T^{2} \)
41 \( 1 + 12.5T + 41T^{2} \)
43 \( 1 - 9.72T + 43T^{2} \)
47 \( 1 - 10.0T + 47T^{2} \)
53 \( 1 - 5.02T + 53T^{2} \)
59 \( 1 - 8.18T + 59T^{2} \)
61 \( 1 + 10.5T + 61T^{2} \)
67 \( 1 - 7.01T + 67T^{2} \)
71 \( 1 - 6.60T + 71T^{2} \)
73 \( 1 + 11.4T + 73T^{2} \)
79 \( 1 - 16.4T + 79T^{2} \)
83 \( 1 + 0.738T + 83T^{2} \)
89 \( 1 - 14.9T + 89T^{2} \)
97 \( 1 + 8.34T + 97T^{2} \)
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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

−8.840841030047948178152666135568, −7.926131527709920133250807900903, −7.00095915346307347874533296578, −6.68073330448483660152419464055, −5.85911099940280088656921831247, −4.58886767527978354560720684388, −3.60493657760467055378609416535, −2.46496760885504086197350241653, −1.82266783925318096960766586476, −0.821671387802816714286149828896, 0.821671387802816714286149828896, 1.82266783925318096960766586476, 2.46496760885504086197350241653, 3.60493657760467055378609416535, 4.58886767527978354560720684388, 5.85911099940280088656921831247, 6.68073330448483660152419464055, 7.00095915346307347874533296578, 7.926131527709920133250807900903, 8.840841030047948178152666135568

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