Properties

Label 2-1014-1.1-c1-0-1
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 − 2.13·5-s + 6-s + 0.0489·7-s − 8-s + 9-s + 2.13·10-s − 6.29·11-s − 12-s − 0.0489·14-s + 2.13·15-s + 16-s − 2.89·17-s − 18-s + 7.20·19-s − 2.13·20-s − 0.0489·21-s + 6.29·22-s + 2.71·23-s + 24-s − 0.432·25-s − 27-s + 0.0489·28-s + 4.91·29-s − 2.13·30-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.955·5-s + 0.408·6-s + 0.0184·7-s − 0.353·8-s + 0.333·9-s + 0.675·10-s − 1.89·11-s − 0.288·12-s − 0.0130·14-s + 0.551·15-s + 0.250·16-s − 0.700·17-s − 0.235·18-s + 1.65·19-s − 0.477·20-s − 0.0106·21-s + 1.34·22-s + 0.565·23-s + 0.204·24-s − 0.0865·25-s − 0.192·27-s + 0.00924·28-s + 0.912·29-s − 0.390·30-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: \(0\)
Selberg data: \((2,\ 1014,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5406647362\)
\(L(\frac12)\) \(\approx\) \(0.5406647362\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.13T + 5T^{2} \)
7 \( 1 - 0.0489T + 7T^{2} \)
11 \( 1 + 6.29T + 11T^{2} \)
17 \( 1 + 2.89T + 17T^{2} \)
19 \( 1 - 7.20T + 19T^{2} \)
23 \( 1 - 2.71T + 23T^{2} \)
29 \( 1 - 4.91T + 29T^{2} \)
31 \( 1 + 9.00T + 31T^{2} \)
37 \( 1 - 0.176T + 37T^{2} \)
41 \( 1 - 8.59T + 41T^{2} \)
43 \( 1 - 6.71T + 43T^{2} \)
47 \( 1 + 7.20T + 47T^{2} \)
53 \( 1 - 9.34T + 53T^{2} \)
59 \( 1 - 4.26T + 59T^{2} \)
61 \( 1 - 7.10T + 61T^{2} \)
67 \( 1 + 5.38T + 67T^{2} \)
71 \( 1 - 8.71T + 71T^{2} \)
73 \( 1 - 14.9T + 73T^{2} \)
79 \( 1 - 13.8T + 79T^{2} \)
83 \( 1 - 11.1T + 83T^{2} \)
89 \( 1 + 3.92T + 89T^{2} \)
97 \( 1 - 2.47T + 97T^{2} \)
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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

−10.00390947512864065102616632136, −9.186457851045932466113364521095, −8.046935348119268823976898860064, −7.63854300222723316824918697266, −6.85551515232118287896093746738, −5.58602573002298444695662775525, −4.90864646768007056794627738194, −3.57421431651013963229120400384, −2.42675244976133488697692612131, −0.62725804158590284016844894758, 0.62725804158590284016844894758, 2.42675244976133488697692612131, 3.57421431651013963229120400384, 4.90864646768007056794627738194, 5.58602573002298444695662775525, 6.85551515232118287896093746738, 7.63854300222723316824918697266, 8.046935348119268823976898860064, 9.186457851045932466113364521095, 10.00390947512864065102616632136

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