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Properties

Label 1-2005-2005.42-r0-0-0
Degree $1$
Conductor $2005$
Sign $0.942 + 0.335i$
Analytic cond. $9.31118$
Root an. cond. $9.31118$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.975 + 0.218i)2-s + (0.985 − 0.171i)3-s + (0.904 + 0.425i)4-s + (0.998 + 0.0471i)6-s + (0.960 + 0.278i)7-s + (0.790 + 0.612i)8-s + (0.940 − 0.338i)9-s + (0.509 + 0.860i)11-s + (0.964 + 0.263i)12-s + (−0.140 − 0.990i)13-s + (0.876 + 0.481i)14-s + (0.637 + 0.770i)16-s + (−0.979 + 0.202i)17-s + (0.992 − 0.125i)18-s + (0.109 − 0.993i)19-s + ⋯
L(s)  = 1  + (0.975 + 0.218i)2-s + (0.985 − 0.171i)3-s + (0.904 + 0.425i)4-s + (0.998 + 0.0471i)6-s + (0.960 + 0.278i)7-s + (0.790 + 0.612i)8-s + (0.940 − 0.338i)9-s + (0.509 + 0.860i)11-s + (0.964 + 0.263i)12-s + (−0.140 − 0.990i)13-s + (0.876 + 0.481i)14-s + (0.637 + 0.770i)16-s + (−0.979 + 0.202i)17-s + (0.992 − 0.125i)18-s + (0.109 − 0.993i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2005 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.942 + 0.335i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2005 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.942 + 0.335i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2005\)    =    \(5 \cdot 401\)
Sign: $0.942 + 0.335i$
Analytic conductor: \(9.31118\)
Root analytic conductor: \(9.31118\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2005} (42, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2005,\ (0:\ ),\ 0.942 + 0.335i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(5.306151595 + 0.9168806015i\)
\(L(\frac12)\) \(\approx\) \(5.306151595 + 0.9168806015i\)
\(L(1)\) \(\approx\) \(2.952183404 + 0.3626792762i\)
\(L(1)\) \(\approx\) \(2.952183404 + 0.3626792762i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
401 \( 1 \)
good2 \( 1 + (0.975 + 0.218i)T \)
3 \( 1 + (0.985 - 0.171i)T \)
7 \( 1 + (0.960 + 0.278i)T \)
11 \( 1 + (0.509 + 0.860i)T \)
13 \( 1 + (-0.140 - 0.990i)T \)
17 \( 1 + (-0.979 + 0.202i)T \)
19 \( 1 + (0.109 - 0.993i)T \)
23 \( 1 + (-0.999 + 0.0157i)T \)
29 \( 1 + (-0.809 - 0.587i)T \)
31 \( 1 + (0.353 - 0.935i)T \)
37 \( 1 + (0.835 + 0.549i)T \)
41 \( 1 + (0.992 - 0.125i)T \)
43 \( 1 + (-0.750 + 0.661i)T \)
47 \( 1 + (0.0314 - 0.999i)T \)
53 \( 1 + (0.625 + 0.780i)T \)
59 \( 1 + (-0.999 + 0.0157i)T \)
61 \( 1 + (0.979 - 0.202i)T \)
67 \( 1 + (0.0471 + 0.998i)T \)
71 \( 1 + (-0.600 + 0.799i)T \)
73 \( 1 + (-0.860 - 0.509i)T \)
79 \( 1 + (-0.467 + 0.883i)T \)
83 \( 1 + (-0.587 - 0.809i)T \)
89 \( 1 + (-0.940 - 0.338i)T \)
97 \( 1 + (0.549 + 0.835i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−20.08632029614314571788615990670, −19.44298071024050895222921775321, −18.72443294074316704143220384305, −17.86524910038445268055871071104, −16.55084596678468982456426171304, −16.23164772498005919130995278444, −15.253068998479253620369260131115, −14.44051912314336986258527347955, −14.15103233144273973423514121623, −13.60087843062581526933597521080, −12.67317478949029677043072542682, −11.78808395835433186437708854366, −11.144971861560154052556972599450, −10.42019423529300292627853878378, −9.45681864924113294882994288063, −8.6442800122565962912095302051, −7.81899997248571725732752810420, −7.051683886414237498108638985865, −6.19673819443493915788864973639, −5.18429479857076242065107271671, −4.204692918544713029203217071981, −3.94897649704059678987942106997, −2.87734563455412635938542829000, −1.924739949099932292022863368917, −1.389794389010733068291889970, 1.37826863001712697953101193744, 2.28991913796329012840934160610, 2.731382441342866661413622048797, 4.10673064143687642055460666211, 4.350121236074616775405263754074, 5.39516044590932984514804509459, 6.343984340623665983564219274437, 7.24408135794806517061054618991, 7.8338675672972346446193097730, 8.49875774683158822933993665615, 9.472160482894831304285382851729, 10.38279699788554150884107872829, 11.40499937691807843603102458931, 11.95618692804663855180651305531, 12.985987133159581830370546332, 13.30390129160698533558028616650, 14.22596934233518780279633778170, 14.96471300433045897354336729869, 15.19301451201731691517218537575, 15.91612443969575381076908036755, 17.200365701918763801446594572817, 17.69840137040362634172461396721, 18.51242556443052407257571702480, 19.713404630656147018922045618440, 20.16825723583532168826104980388

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