Properties

Label 2-4031-1.1-c1-0-236
Degree $2$
Conductor $4031$
Sign $-1$
Analytic cond. $32.1876$
Root an. cond. $5.67342$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.44·2-s + 1.49·3-s + 0.0743·4-s + 2.03·5-s − 2.15·6-s − 3.27·7-s + 2.77·8-s − 0.766·9-s − 2.93·10-s + 2.62·11-s + 0.111·12-s − 5.46·13-s + 4.71·14-s + 3.04·15-s − 4.14·16-s + 4.75·17-s + 1.10·18-s + 3.60·19-s + 0.151·20-s − 4.89·21-s − 3.77·22-s − 2.87·23-s + 4.14·24-s − 0.843·25-s + 7.87·26-s − 5.62·27-s − 0.243·28-s + ⋯
L(s)  = 1  − 1.01·2-s + 0.862·3-s + 0.0371·4-s + 0.911·5-s − 0.878·6-s − 1.23·7-s + 0.980·8-s − 0.255·9-s − 0.928·10-s + 0.790·11-s + 0.0320·12-s − 1.51·13-s + 1.26·14-s + 0.786·15-s − 1.03·16-s + 1.15·17-s + 0.260·18-s + 0.828·19-s + 0.0338·20-s − 1.06·21-s − 0.804·22-s − 0.599·23-s + 0.846·24-s − 0.168·25-s + 1.54·26-s − 1.08·27-s − 0.0460·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4031 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4031 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4031\)    =    \(29 \cdot 139\)
Sign: $-1$
Analytic conductor: \(32.1876\)
Root analytic conductor: \(5.67342\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 4031,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad29 \( 1 + T \)
139 \( 1 + T \)
good2 \( 1 + 1.44T + 2T^{2} \)
3 \( 1 - 1.49T + 3T^{2} \)
5 \( 1 - 2.03T + 5T^{2} \)
7 \( 1 + 3.27T + 7T^{2} \)
11 \( 1 - 2.62T + 11T^{2} \)
13 \( 1 + 5.46T + 13T^{2} \)
17 \( 1 - 4.75T + 17T^{2} \)
19 \( 1 - 3.60T + 19T^{2} \)
23 \( 1 + 2.87T + 23T^{2} \)
31 \( 1 - 1.86T + 31T^{2} \)
37 \( 1 - 9.24T + 37T^{2} \)
41 \( 1 + 7.59T + 41T^{2} \)
43 \( 1 + 2.51T + 43T^{2} \)
47 \( 1 + 0.622T + 47T^{2} \)
53 \( 1 - 3.25T + 53T^{2} \)
59 \( 1 + 1.92T + 59T^{2} \)
61 \( 1 - 0.556T + 61T^{2} \)
67 \( 1 + 8.15T + 67T^{2} \)
71 \( 1 + 0.307T + 71T^{2} \)
73 \( 1 - 4.83T + 73T^{2} \)
79 \( 1 - 0.0396T + 79T^{2} \)
83 \( 1 + 11.7T + 83T^{2} \)
89 \( 1 + 4.78T + 89T^{2} \)
97 \( 1 + 13.7T + 97T^{2} \)
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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

−8.151245409763813915641815723910, −7.58848028141323822927557657434, −6.84471627081383859561229579524, −5.96179810347037054422006013147, −5.21454969661137095156198108005, −4.05916959521118295907990985503, −3.14465234438588797695992573456, −2.40513184437539781282479818672, −1.37585407445402730088725277218, 0, 1.37585407445402730088725277218, 2.40513184437539781282479818672, 3.14465234438588797695992573456, 4.05916959521118295907990985503, 5.21454969661137095156198108005, 5.96179810347037054422006013147, 6.84471627081383859561229579524, 7.58848028141323822927557657434, 8.151245409763813915641815723910

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