Properties

Label 2-1003-1.1-c1-0-43
Degree $2$
Conductor $1003$
Sign $1$
Analytic cond. $8.00899$
Root an. cond. $2.83001$
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  + 2.73·2-s − 0.137·3-s + 5.47·4-s − 1.85·5-s − 0.375·6-s + 0.325·7-s + 9.50·8-s − 2.98·9-s − 5.06·10-s + 4.12·11-s − 0.752·12-s + 4.56·13-s + 0.888·14-s + 0.254·15-s + 15.0·16-s − 17-s − 8.15·18-s + 4.92·19-s − 10.1·20-s − 0.0446·21-s + 11.2·22-s − 5.18·23-s − 1.30·24-s − 1.56·25-s + 12.4·26-s + 0.821·27-s + 1.78·28-s + ⋯
L(s)  = 1  + 1.93·2-s − 0.0793·3-s + 2.73·4-s − 0.828·5-s − 0.153·6-s + 0.122·7-s + 3.35·8-s − 0.993·9-s − 1.60·10-s + 1.24·11-s − 0.217·12-s + 1.26·13-s + 0.237·14-s + 0.0657·15-s + 3.75·16-s − 0.242·17-s − 1.92·18-s + 1.13·19-s − 2.26·20-s − 0.00974·21-s + 2.40·22-s − 1.08·23-s − 0.266·24-s − 0.313·25-s + 2.44·26-s + 0.158·27-s + 0.336·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1003 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1003 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1003\)    =    \(17 \cdot 59\)
Sign: $1$
Analytic conductor: \(8.00899\)
Root analytic conductor: \(2.83001\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1003,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.803160479\)
\(L(\frac12)\) \(\approx\) \(4.803160479\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 + T \)
59 \( 1 - T \)
good2 \( 1 - 2.73T + 2T^{2} \)
3 \( 1 + 0.137T + 3T^{2} \)
5 \( 1 + 1.85T + 5T^{2} \)
7 \( 1 - 0.325T + 7T^{2} \)
11 \( 1 - 4.12T + 11T^{2} \)
13 \( 1 - 4.56T + 13T^{2} \)
19 \( 1 - 4.92T + 19T^{2} \)
23 \( 1 + 5.18T + 23T^{2} \)
29 \( 1 + 6.24T + 29T^{2} \)
31 \( 1 - 2.75T + 31T^{2} \)
37 \( 1 + 4.40T + 37T^{2} \)
41 \( 1 - 3.43T + 41T^{2} \)
43 \( 1 + 8.65T + 43T^{2} \)
47 \( 1 - 12.5T + 47T^{2} \)
53 \( 1 - 0.346T + 53T^{2} \)
61 \( 1 + 15.1T + 61T^{2} \)
67 \( 1 + 13.1T + 67T^{2} \)
71 \( 1 - 2.72T + 71T^{2} \)
73 \( 1 + 11.0T + 73T^{2} \)
79 \( 1 + 5.84T + 79T^{2} \)
83 \( 1 + 5.47T + 83T^{2} \)
89 \( 1 - 0.986T + 89T^{2} \)
97 \( 1 - 0.328T + 97T^{2} \)
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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

−10.46527194659563753508319182785, −9.020645753469281550918458695523, −7.980425921781630760533317538825, −7.17183158861522234928471315581, −6.12706737142108097784746687214, −5.74249302661817703229764405104, −4.51214634664209440729686094739, −3.75528004823199542073060463620, −3.16512440474011427721789398826, −1.64248954246910494096196689379, 1.64248954246910494096196689379, 3.16512440474011427721789398826, 3.75528004823199542073060463620, 4.51214634664209440729686094739, 5.74249302661817703229764405104, 6.12706737142108097784746687214, 7.17183158861522234928471315581, 7.980425921781630760533317538825, 9.020645753469281550918458695523, 10.46527194659563753508319182785

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