Properties

Label 2-1003-1.1-c1-0-26
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  + 1.49·2-s − 1.13·3-s + 0.240·4-s + 3.48·5-s − 1.69·6-s − 1.05·7-s − 2.63·8-s − 1.71·9-s + 5.21·10-s + 5.79·11-s − 0.272·12-s + 2.73·13-s − 1.58·14-s − 3.95·15-s − 4.42·16-s − 17-s − 2.55·18-s + 4.97·19-s + 0.836·20-s + 1.19·21-s + 8.67·22-s + 3.90·23-s + 2.99·24-s + 7.11·25-s + 4.09·26-s + 5.34·27-s − 0.253·28-s + ⋯
L(s)  = 1  + 1.05·2-s − 0.655·3-s + 0.120·4-s + 1.55·5-s − 0.693·6-s − 0.399·7-s − 0.931·8-s − 0.570·9-s + 1.64·10-s + 1.74·11-s − 0.0787·12-s + 0.759·13-s − 0.422·14-s − 1.02·15-s − 1.10·16-s − 0.242·17-s − 0.603·18-s + 1.14·19-s + 0.187·20-s + 0.261·21-s + 1.84·22-s + 0.814·23-s + 0.610·24-s + 1.42·25-s + 0.804·26-s + 1.02·27-s − 0.0479·28-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

\(L(1)\) \(\approx\) \(2.604049770\)
\(L(\frac12)\) \(\approx\) \(2.604049770\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.49T + 2T^{2} \)
3 \( 1 + 1.13T + 3T^{2} \)
5 \( 1 - 3.48T + 5T^{2} \)
7 \( 1 + 1.05T + 7T^{2} \)
11 \( 1 - 5.79T + 11T^{2} \)
13 \( 1 - 2.73T + 13T^{2} \)
19 \( 1 - 4.97T + 19T^{2} \)
23 \( 1 - 3.90T + 23T^{2} \)
29 \( 1 - 0.0187T + 29T^{2} \)
31 \( 1 + 5.26T + 31T^{2} \)
37 \( 1 - 10.5T + 37T^{2} \)
41 \( 1 - 11.1T + 41T^{2} \)
43 \( 1 + 12.9T + 43T^{2} \)
47 \( 1 + 6.43T + 47T^{2} \)
53 \( 1 - 3.67T + 53T^{2} \)
61 \( 1 + 4.10T + 61T^{2} \)
67 \( 1 - 12.3T + 67T^{2} \)
71 \( 1 - 0.517T + 71T^{2} \)
73 \( 1 - 10.8T + 73T^{2} \)
79 \( 1 + 12.1T + 79T^{2} \)
83 \( 1 + 13.5T + 83T^{2} \)
89 \( 1 + 6.25T + 89T^{2} \)
97 \( 1 - 2.06T + 97T^{2} \)
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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

−9.722026497237046914805258011207, −9.385704224698776738011074519957, −8.604088598153754256149716347661, −6.77404029427285267861193751652, −6.25565557799727912491276167634, −5.70356386017617261725151422820, −4.95218957437531948908504735048, −3.77383312388777461739416007636, −2.80334750711685515540162806082, −1.24279429723016255679406153848, 1.24279429723016255679406153848, 2.80334750711685515540162806082, 3.77383312388777461739416007636, 4.95218957437531948908504735048, 5.70356386017617261725151422820, 6.25565557799727912491276167634, 6.77404029427285267861193751652, 8.604088598153754256149716347661, 9.385704224698776738011074519957, 9.722026497237046914805258011207

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