Properties

Label 2-1014-1.1-c1-0-23
Degree $2$
Conductor $1014$
Sign $-1$
Analytic cond. $8.09683$
Root an. cond. $2.84549$
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 + 3-s + 4-s − 0.267·5-s − 6-s − 0.732·7-s − 8-s + 9-s + 0.267·10-s − 4.73·11-s + 12-s + 0.732·14-s − 0.267·15-s + 16-s − 2.26·17-s − 18-s − 1.26·19-s − 0.267·20-s − 0.732·21-s + 4.73·22-s − 6.19·23-s − 24-s − 4.92·25-s + 27-s − 0.732·28-s + 2.46·29-s + 0.267·30-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.119·5-s − 0.408·6-s − 0.276·7-s − 0.353·8-s + 0.333·9-s + 0.0847·10-s − 1.42·11-s + 0.288·12-s + 0.195·14-s − 0.0691·15-s + 0.250·16-s − 0.550·17-s − 0.235·18-s − 0.290·19-s − 0.0599·20-s − 0.159·21-s + 1.00·22-s − 1.29·23-s − 0.204·24-s − 0.985·25-s + 0.192·27-s − 0.138·28-s + 0.457·29-s + 0.0489·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1014\)    =    \(2 \cdot 3 \cdot 13^{2}\)
Sign: $-1$
Analytic conductor: \(8.09683\)
Root analytic conductor: \(2.84549\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1014,\ (\ :1/2),\ -1)\)

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 \( 1 + T \)
3 \( 1 - T \)
13 \( 1 \)
good5 \( 1 + 0.267T + 5T^{2} \)
7 \( 1 + 0.732T + 7T^{2} \)
11 \( 1 + 4.73T + 11T^{2} \)
17 \( 1 + 2.26T + 17T^{2} \)
19 \( 1 + 1.26T + 19T^{2} \)
23 \( 1 + 6.19T + 23T^{2} \)
29 \( 1 - 2.46T + 29T^{2} \)
31 \( 1 - 5.46T + 31T^{2} \)
37 \( 1 + 10.4T + 37T^{2} \)
41 \( 1 + 11.3T + 41T^{2} \)
43 \( 1 - 7.66T + 43T^{2} \)
47 \( 1 - 8.19T + 47T^{2} \)
53 \( 1 - 0.464T + 53T^{2} \)
59 \( 1 + 8T + 59T^{2} \)
61 \( 1 - 1.19T + 61T^{2} \)
67 \( 1 - 11.1T + 67T^{2} \)
71 \( 1 + 1.26T + 71T^{2} \)
73 \( 1 - 9.73T + 73T^{2} \)
79 \( 1 + 9.46T + 79T^{2} \)
83 \( 1 + 10.1T + 83T^{2} \)
89 \( 1 - 2.53T + 89T^{2} \)
97 \( 1 - 6T + 97T^{2} \)
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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

−9.623561960004761021141007876597, −8.512179029874987613541000644532, −8.117228014943947903659384821385, −7.24874596078998473462173140795, −6.32910329425655374720158433612, −5.25550663995794891926432539900, −4.02501348995932098996250668558, −2.86416185658516695923656538755, −1.95201350757735839940759885672, 0, 1.95201350757735839940759885672, 2.86416185658516695923656538755, 4.02501348995932098996250668558, 5.25550663995794891926432539900, 6.32910329425655374720158433612, 7.24874596078998473462173140795, 8.117228014943947903659384821385, 8.512179029874987613541000644532, 9.623561960004761021141007876597

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