Properties

Label 2-1003-1.1-c1-0-40
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.85·2-s + 1.40·3-s + 1.43·4-s − 4.08·5-s − 2.60·6-s + 1.71·7-s + 1.04·8-s − 1.02·9-s + 7.57·10-s + 5.46·11-s + 2.01·12-s − 5.16·13-s − 3.17·14-s − 5.74·15-s − 4.81·16-s − 17-s + 1.89·18-s + 6.66·19-s − 5.86·20-s + 2.40·21-s − 10.1·22-s − 6.04·23-s + 1.47·24-s + 11.7·25-s + 9.57·26-s − 5.65·27-s + 2.45·28-s + ⋯
L(s)  = 1  − 1.31·2-s + 0.811·3-s + 0.717·4-s − 1.82·5-s − 1.06·6-s + 0.647·7-s + 0.370·8-s − 0.340·9-s + 2.39·10-s + 1.64·11-s + 0.582·12-s − 1.43·13-s − 0.848·14-s − 1.48·15-s − 1.20·16-s − 0.242·17-s + 0.446·18-s + 1.52·19-s − 1.31·20-s + 0.525·21-s − 2.15·22-s − 1.26·23-s + 0.300·24-s + 2.34·25-s + 1.87·26-s − 1.08·27-s + 0.464·28-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: \(1\)
Selberg data: \((2,\ 1003,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.85T + 2T^{2} \)
3 \( 1 - 1.40T + 3T^{2} \)
5 \( 1 + 4.08T + 5T^{2} \)
7 \( 1 - 1.71T + 7T^{2} \)
11 \( 1 - 5.46T + 11T^{2} \)
13 \( 1 + 5.16T + 13T^{2} \)
19 \( 1 - 6.66T + 19T^{2} \)
23 \( 1 + 6.04T + 23T^{2} \)
29 \( 1 + 1.29T + 29T^{2} \)
31 \( 1 - 7.77T + 31T^{2} \)
37 \( 1 + 2.94T + 37T^{2} \)
41 \( 1 + 8.37T + 41T^{2} \)
43 \( 1 + 4.55T + 43T^{2} \)
47 \( 1 + 3.75T + 47T^{2} \)
53 \( 1 + 9.86T + 53T^{2} \)
61 \( 1 - 3.09T + 61T^{2} \)
67 \( 1 - 6.99T + 67T^{2} \)
71 \( 1 + 9.77T + 71T^{2} \)
73 \( 1 + 8.11T + 73T^{2} \)
79 \( 1 + 3.50T + 79T^{2} \)
83 \( 1 + 6.36T + 83T^{2} \)
89 \( 1 + 5.97T + 89T^{2} \)
97 \( 1 - 6.81T + 97T^{2} \)
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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

−9.417464883189236214680905997807, −8.504251107269394191245160048468, −8.126536725006030564753987382815, −7.47912470424065892179383114254, −6.78247662427744509695873702639, −4.88446824946440935317693635711, −4.06490121626915188588328194355, −3.05874353313088261646516539285, −1.53627026630342604577470159166, 0, 1.53627026630342604577470159166, 3.05874353313088261646516539285, 4.06490121626915188588328194355, 4.88446824946440935317693635711, 6.78247662427744509695873702639, 7.47912470424065892179383114254, 8.126536725006030564753987382815, 8.504251107269394191245160048468, 9.417464883189236214680905997807

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