Properties

Label 2-1003-1.1-c1-0-70
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  + 2.64·2-s + 2.89·3-s + 5.00·4-s − 0.0576·5-s + 7.67·6-s − 4.69·7-s + 7.94·8-s + 5.40·9-s − 0.152·10-s − 2.89·11-s + 14.5·12-s + 1.05·13-s − 12.4·14-s − 0.167·15-s + 11.0·16-s − 17-s + 14.3·18-s − 6.13·19-s − 0.288·20-s − 13.6·21-s − 7.67·22-s − 1.09·23-s + 23.0·24-s − 4.99·25-s + 2.79·26-s + 6.98·27-s − 23.4·28-s + ⋯
L(s)  = 1  + 1.87·2-s + 1.67·3-s + 2.50·4-s − 0.0257·5-s + 3.13·6-s − 1.77·7-s + 2.80·8-s + 1.80·9-s − 0.0482·10-s − 0.874·11-s + 4.18·12-s + 0.292·13-s − 3.32·14-s − 0.0431·15-s + 2.75·16-s − 0.242·17-s + 3.37·18-s − 1.40·19-s − 0.0644·20-s − 2.97·21-s − 1.63·22-s − 0.227·23-s + 4.70·24-s − 0.999·25-s + 0.547·26-s + 1.34·27-s − 4.44·28-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

\(L(1)\) \(\approx\) \(6.784988579\)
\(L(\frac12)\) \(\approx\) \(6.784988579\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.64T + 2T^{2} \)
3 \( 1 - 2.89T + 3T^{2} \)
5 \( 1 + 0.0576T + 5T^{2} \)
7 \( 1 + 4.69T + 7T^{2} \)
11 \( 1 + 2.89T + 11T^{2} \)
13 \( 1 - 1.05T + 13T^{2} \)
19 \( 1 + 6.13T + 19T^{2} \)
23 \( 1 + 1.09T + 23T^{2} \)
29 \( 1 + 3.98T + 29T^{2} \)
31 \( 1 - 6.59T + 31T^{2} \)
37 \( 1 - 3.90T + 37T^{2} \)
41 \( 1 - 0.303T + 41T^{2} \)
43 \( 1 - 5.89T + 43T^{2} \)
47 \( 1 - 9.89T + 47T^{2} \)
53 \( 1 - 14.1T + 53T^{2} \)
61 \( 1 + 3.42T + 61T^{2} \)
67 \( 1 - 5.37T + 67T^{2} \)
71 \( 1 - 3.29T + 71T^{2} \)
73 \( 1 - 7.44T + 73T^{2} \)
79 \( 1 + 7.79T + 79T^{2} \)
83 \( 1 + 9.61T + 83T^{2} \)
89 \( 1 - 16.8T + 89T^{2} \)
97 \( 1 + 4.46T + 97T^{2} \)
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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

−10.06744087119611553858677753843, −9.155738383445901891247788690691, −8.099029425520928322393418407720, −7.23899683334419719504634478517, −6.44186052612664536284234866725, −5.66945571554876511789975186119, −4.13441317476856254798452009440, −3.80455858295328692776650408158, −2.70143569036161562639270824154, −2.36322602006988945780959734512, 2.36322602006988945780959734512, 2.70143569036161562639270824154, 3.80455858295328692776650408158, 4.13441317476856254798452009440, 5.66945571554876511789975186119, 6.44186052612664536284234866725, 7.23899683334419719504634478517, 8.099029425520928322393418407720, 9.155738383445901891247788690691, 10.06744087119611553858677753843

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