Properties

Label 2-1014-1.1-c1-0-9
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 − 4.04·5-s + 6-s − 0.692·7-s + 8-s + 9-s − 4.04·10-s + 4.85·11-s + 12-s − 0.692·14-s − 4.04·15-s + 16-s + 7.38·17-s + 18-s + 1.78·19-s − 4.04·20-s − 0.692·21-s + 4.85·22-s + 5.10·23-s + 24-s + 11.3·25-s + 27-s − 0.692·28-s − 3.34·29-s − 4.04·30-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 0.5·4-s − 1.81·5-s + 0.408·6-s − 0.261·7-s + 0.353·8-s + 0.333·9-s − 1.28·10-s + 1.46·11-s + 0.288·12-s − 0.184·14-s − 1.04·15-s + 0.250·16-s + 1.79·17-s + 0.235·18-s + 0.408·19-s − 0.905·20-s − 0.151·21-s + 1.03·22-s + 1.06·23-s + 0.204·24-s + 2.27·25-s + 0.192·27-s − 0.130·28-s − 0.621·29-s − 0.739·30-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

\(L(1)\) \(\approx\) \(2.470799816\)
\(L(\frac12)\) \(\approx\) \(2.470799816\)
\(L(\frac{3}{2})\) 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 \( 1 + 4.04T + 5T^{2} \)
7 \( 1 + 0.692T + 7T^{2} \)
11 \( 1 - 4.85T + 11T^{2} \)
17 \( 1 - 7.38T + 17T^{2} \)
19 \( 1 - 1.78T + 19T^{2} \)
23 \( 1 - 5.10T + 23T^{2} \)
29 \( 1 + 3.34T + 29T^{2} \)
31 \( 1 + 0.972T + 31T^{2} \)
37 \( 1 + 1.28T + 37T^{2} \)
41 \( 1 + 1.50T + 41T^{2} \)
43 \( 1 + 8.31T + 43T^{2} \)
47 \( 1 - 7.20T + 47T^{2} \)
53 \( 1 - 13.4T + 53T^{2} \)
59 \( 1 + 1.30T + 59T^{2} \)
61 \( 1 + 0.396T + 61T^{2} \)
67 \( 1 + 6.05T + 67T^{2} \)
71 \( 1 - 1.32T + 71T^{2} \)
73 \( 1 - 7.65T + 73T^{2} \)
79 \( 1 + 8.33T + 79T^{2} \)
83 \( 1 + 15.3T + 83T^{2} \)
89 \( 1 + 3.10T + 89T^{2} \)
97 \( 1 - 8.54T + 97T^{2} \)
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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

−9.956414029867099092408017800636, −8.979556125130113988977573990488, −8.175257530052979246896455844341, −7.32414158715904383617297145133, −6.86070391608869888729693546101, −5.49796056869188660020166585707, −4.36982047888807898836709379631, −3.60348136083605367309432240033, −3.14822827482958216267996048640, −1.18236672390134377279201613725, 1.18236672390134377279201613725, 3.14822827482958216267996048640, 3.60348136083605367309432240033, 4.36982047888807898836709379631, 5.49796056869188660020166585707, 6.86070391608869888729693546101, 7.32414158715904383617297145133, 8.175257530052979246896455844341, 8.979556125130113988977573990488, 9.956414029867099092408017800636

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