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Properties

Label 1-1003-1003.331-r0-0-0
Degree $1$
Conductor $1003$
Sign $0.989 + 0.143i$
Analytic cond. $4.65791$
Root an. cond. $4.65791$
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  + (−0.998 − 0.0541i)2-s + (−0.938 − 0.344i)3-s + (0.994 + 0.108i)4-s + (−0.444 − 0.895i)5-s + (0.918 + 0.395i)6-s + (−0.982 − 0.188i)7-s + (−0.986 − 0.161i)8-s + (0.762 + 0.647i)9-s + (0.395 + 0.918i)10-s + (−0.538 + 0.842i)11-s + (−0.895 − 0.444i)12-s + (−0.647 − 0.762i)13-s + (0.970 + 0.241i)14-s + (0.108 + 0.994i)15-s + (0.976 + 0.214i)16-s + ⋯
L(s)  = 1  + (−0.998 − 0.0541i)2-s + (−0.938 − 0.344i)3-s + (0.994 + 0.108i)4-s + (−0.444 − 0.895i)5-s + (0.918 + 0.395i)6-s + (−0.982 − 0.188i)7-s + (−0.986 − 0.161i)8-s + (0.762 + 0.647i)9-s + (0.395 + 0.918i)10-s + (−0.538 + 0.842i)11-s + (−0.895 − 0.444i)12-s + (−0.647 − 0.762i)13-s + (0.970 + 0.241i)14-s + (0.108 + 0.994i)15-s + (0.976 + 0.214i)16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1003 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.989 + 0.143i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1003 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.989 + 0.143i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1003\)    =    \(17 \cdot 59\)
Sign: $0.989 + 0.143i$
Analytic conductor: \(4.65791\)
Root analytic conductor: \(4.65791\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1003} (331, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1003,\ (0:\ ),\ 0.989 + 0.143i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1836203063 + 0.01328258653i\)
\(L(\frac12)\) \(\approx\) \(0.1836203063 + 0.01328258653i\)
\(L(1)\) \(\approx\) \(0.3227304681 - 0.09514168950i\)
\(L(1)\) \(\approx\) \(0.3227304681 - 0.09514168950i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 \)
59 \( 1 \)
good2 \( 1 + (-0.998 - 0.0541i)T \)
3 \( 1 + (-0.938 - 0.344i)T \)
5 \( 1 + (-0.444 - 0.895i)T \)
7 \( 1 + (-0.982 - 0.188i)T \)
11 \( 1 + (-0.538 + 0.842i)T \)
13 \( 1 + (-0.647 - 0.762i)T \)
19 \( 1 + (-0.883 - 0.468i)T \)
23 \( 1 + (0.0270 - 0.999i)T \)
29 \( 1 + (-0.667 + 0.744i)T \)
31 \( 1 + (0.293 - 0.955i)T \)
37 \( 1 + (-0.583 + 0.812i)T \)
41 \( 1 + (-0.999 + 0.0270i)T \)
43 \( 1 + (-0.214 + 0.976i)T \)
47 \( 1 + (0.947 - 0.319i)T \)
53 \( 1 + (-0.928 + 0.370i)T \)
61 \( 1 + (-0.667 - 0.744i)T \)
67 \( 1 + (-0.161 + 0.986i)T \)
71 \( 1 + (0.895 + 0.444i)T \)
73 \( 1 + (-0.970 - 0.241i)T \)
79 \( 1 + (-0.938 + 0.344i)T \)
83 \( 1 + (0.963 - 0.267i)T \)
89 \( 1 + (-0.0541 - 0.998i)T \)
97 \( 1 + (-0.241 - 0.970i)T \)
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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

−21.65754062324712231440199631910, −20.9888470669252854860775574508, −19.68480764536542226146752011640, −18.9386600711094686159883572795, −18.75726211189029803893858371429, −17.67164926314228516209713054431, −16.921834256611703787049412727136, −16.22775067161284461648529926150, −15.58938264092591738994786793162, −14.99364364008372682309354085196, −13.75522396895148112253408365440, −12.42356190928161025300456349005, −11.85367507737835422816794191890, −10.98433803771371985537025992860, −10.42260481076511148124343364452, −9.70512310940687453934973645873, −8.83663286647379854292361152760, −7.63561093586804630742477464933, −6.87192250851735340987826087864, −6.23495543113112762048944860081, −5.447945857604424316997949355556, −3.891180291368443259734364251362, −3.09831407306446931406856050537, −1.93868325933897740322926848363, −0.238729042926728566137236133419, 0.53899025578559061750106583865, 1.78034076372923476699452417066, 2.88465291223757842656416616738, 4.32484448592197246301969952068, 5.24561906416884307664503199072, 6.26171156504775532075467144788, 7.08479447966494223250709037168, 7.75551886984921439156359515535, 8.68430691456815156742884559204, 9.75996623342330555485605368052, 10.28591220189481777427596485740, 11.159704613551039018624638820292, 12.20185435757879475243428317496, 12.64201773458553310837308180986, 13.19471919961048939597382856294, 15.093970199927293045115573517360, 15.61903892898351356713497760880, 16.47189616630097496142829213125, 17.02450437453781205952269949210, 17.55772026783162757994387072606, 18.628225108935686802736085236980, 19.12065091773526710663149710619, 20.14656106805543537472482156918, 20.44182276038401018820490452272, 21.66916177226794733408952544003

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