Properties

Label 2-1014-1.1-c1-0-2
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 3-s + 4-s − 0.356·5-s − 6-s − 4.04·7-s − 8-s + 9-s + 0.356·10-s + 0.911·11-s + 12-s + 4.04·14-s − 0.356·15-s + 16-s − 2.09·17-s − 18-s + 4.98·19-s − 0.356·20-s − 4.04·21-s − 0.911·22-s + 8.49·23-s − 24-s − 4.87·25-s + 27-s − 4.04·28-s + 8.51·29-s + 0.356·30-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.159·5-s − 0.408·6-s − 1.53·7-s − 0.353·8-s + 0.333·9-s + 0.112·10-s + 0.274·11-s + 0.288·12-s + 1.08·14-s − 0.0921·15-s + 0.250·16-s − 0.508·17-s − 0.235·18-s + 1.14·19-s − 0.0798·20-s − 0.883·21-s − 0.194·22-s + 1.77·23-s − 0.204·24-s − 0.974·25-s + 0.192·27-s − 0.765·28-s + 1.58·29-s + 0.0651·30-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: $\chi_{1014} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1014,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.187794774\)
\(L(\frac12)\) \(\approx\) \(1.187794774\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.356T + 5T^{2} \)
7 \( 1 + 4.04T + 7T^{2} \)
11 \( 1 - 0.911T + 11T^{2} \)
17 \( 1 + 2.09T + 17T^{2} \)
19 \( 1 - 4.98T + 19T^{2} \)
23 \( 1 - 8.49T + 23T^{2} \)
29 \( 1 - 8.51T + 29T^{2} \)
31 \( 1 - 10.7T + 31T^{2} \)
37 \( 1 - 0.615T + 37T^{2} \)
41 \( 1 - 7.60T + 41T^{2} \)
43 \( 1 + 6.27T + 43T^{2} \)
47 \( 1 + 1.78T + 47T^{2} \)
53 \( 1 - 10.4T + 53T^{2} \)
59 \( 1 - 6.04T + 59T^{2} \)
61 \( 1 + 3.10T + 61T^{2} \)
67 \( 1 + 13.5T + 67T^{2} \)
71 \( 1 + 11.4T + 71T^{2} \)
73 \( 1 + 0.533T + 73T^{2} \)
79 \( 1 + 11.7T + 79T^{2} \)
83 \( 1 - 6.49T + 83T^{2} \)
89 \( 1 - 6.49T + 89T^{2} \)
97 \( 1 - 1.96T + 97T^{2} \)
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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

−9.870824776878324751252205154115, −9.135311224362448943950807052588, −8.515946485045853002081019226149, −7.43477749687803727867578798690, −6.77834408758778658600133648616, −6.00421437108238013889510175075, −4.55599552114216611354408033010, −3.27963387769484010305639416499, −2.69595240870254514358033529224, −0.925337255659619455615447351878, 0.925337255659619455615447351878, 2.69595240870254514358033529224, 3.27963387769484010305639416499, 4.55599552114216611354408033010, 6.00421437108238013889510175075, 6.77834408758778658600133648616, 7.43477749687803727867578798690, 8.515946485045853002081019226149, 9.135311224362448943950807052588, 9.870824776878324751252205154115

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