Properties

Label 2-4010-1.1-c1-0-94
Degree $2$
Conductor $4010$
Sign $-1$
Analytic cond. $32.0200$
Root an. cond. $5.65862$
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  − 2-s + 0.271·3-s + 4-s − 5-s − 0.271·6-s + 1.75·7-s − 8-s − 2.92·9-s + 10-s + 0.0291·11-s + 0.271·12-s + 5.25·13-s − 1.75·14-s − 0.271·15-s + 16-s − 3.06·17-s + 2.92·18-s − 6.99·19-s − 20-s + 0.477·21-s − 0.0291·22-s + 3.94·23-s − 0.271·24-s + 25-s − 5.25·26-s − 1.60·27-s + 1.75·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.156·3-s + 0.5·4-s − 0.447·5-s − 0.110·6-s + 0.665·7-s − 0.353·8-s − 0.975·9-s + 0.316·10-s + 0.00878·11-s + 0.0783·12-s + 1.45·13-s − 0.470·14-s − 0.0701·15-s + 0.250·16-s − 0.744·17-s + 0.689·18-s − 1.60·19-s − 0.223·20-s + 0.104·21-s − 0.00620·22-s + 0.823·23-s − 0.0554·24-s + 0.200·25-s − 1.03·26-s − 0.309·27-s + 0.332·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4010 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4010 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4010\)    =    \(2 \cdot 5 \cdot 401\)
Sign: $-1$
Analytic conductor: \(32.0200\)
Root analytic conductor: \(5.65862\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 4010,\ (\ :1/2),\ -1)\)

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)$
bad2 \( 1 + T \)
5 \( 1 + T \)
401 \( 1 + T \)
good3 \( 1 - 0.271T + 3T^{2} \)
7 \( 1 - 1.75T + 7T^{2} \)
11 \( 1 - 0.0291T + 11T^{2} \)
13 \( 1 - 5.25T + 13T^{2} \)
17 \( 1 + 3.06T + 17T^{2} \)
19 \( 1 + 6.99T + 19T^{2} \)
23 \( 1 - 3.94T + 23T^{2} \)
29 \( 1 - 8.01T + 29T^{2} \)
31 \( 1 + 7.61T + 31T^{2} \)
37 \( 1 + 3.48T + 37T^{2} \)
41 \( 1 - 1.23T + 41T^{2} \)
43 \( 1 - 3.83T + 43T^{2} \)
47 \( 1 + 11.7T + 47T^{2} \)
53 \( 1 - 13.3T + 53T^{2} \)
59 \( 1 - 1.28T + 59T^{2} \)
61 \( 1 + 5.25T + 61T^{2} \)
67 \( 1 + 2.24T + 67T^{2} \)
71 \( 1 + 11.0T + 71T^{2} \)
73 \( 1 + 15.0T + 73T^{2} \)
79 \( 1 + 2.43T + 79T^{2} \)
83 \( 1 - 8.86T + 83T^{2} \)
89 \( 1 - 12.0T + 89T^{2} \)
97 \( 1 - 11.8T + 97T^{2} \)
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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

−8.332189134220904406137302653508, −7.56329516745542786610030002023, −6.60623707583315976647944706807, −6.11248685454691414789994068140, −5.09509773860663945414231139701, −4.18810977554811713739587018087, −3.30501885960915770936615965949, −2.35117619110972442694153945615, −1.34151703321977202334308445314, 0, 1.34151703321977202334308445314, 2.35117619110972442694153945615, 3.30501885960915770936615965949, 4.18810977554811713739587018087, 5.09509773860663945414231139701, 6.11248685454691414789994068140, 6.60623707583315976647944706807, 7.56329516745542786610030002023, 8.332189134220904406137302653508

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