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Properties

Label 1-2003-2003.180-r1-0-0
Degree $1$
Conductor $2003$
Sign $0.0938 + 0.995i$
Analytic cond. $215.252$
Root an. cond. $215.252$
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  + (−0.841 − 0.540i)2-s + (0.841 − 0.540i)3-s + (0.415 + 0.909i)4-s + (−0.415 − 0.909i)5-s − 6-s + (0.654 − 0.755i)7-s + (0.142 − 0.989i)8-s + (0.415 − 0.909i)9-s + (−0.142 + 0.989i)10-s + (−0.841 + 0.540i)11-s + (0.841 + 0.540i)12-s + (−0.654 − 0.755i)13-s + (−0.959 + 0.281i)14-s + (−0.841 − 0.540i)15-s + (−0.654 + 0.755i)16-s − 17-s + ⋯
L(s)  = 1  + (−0.841 − 0.540i)2-s + (0.841 − 0.540i)3-s + (0.415 + 0.909i)4-s + (−0.415 − 0.909i)5-s − 6-s + (0.654 − 0.755i)7-s + (0.142 − 0.989i)8-s + (0.415 − 0.909i)9-s + (−0.142 + 0.989i)10-s + (−0.841 + 0.540i)11-s + (0.841 + 0.540i)12-s + (−0.654 − 0.755i)13-s + (−0.959 + 0.281i)14-s + (−0.841 − 0.540i)15-s + (−0.654 + 0.755i)16-s − 17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2003 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0938 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2003 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0938 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2003\)
Sign: $0.0938 + 0.995i$
Analytic conductor: \(215.252\)
Root analytic conductor: \(215.252\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2003} (180, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2003,\ (1:\ ),\ 0.0938 + 0.995i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.9385868722 - 0.8542832901i\)
\(L(\frac12)\) \(\approx\) \(-0.9385868722 - 0.8542832901i\)
\(L(1)\) \(\approx\) \(0.4827282442 - 0.6650962209i\)
\(L(1)\) \(\approx\) \(0.4827282442 - 0.6650962209i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2003 \( 1 \)
good2 \( 1 + (-0.841 - 0.540i)T \)
3 \( 1 + (0.841 - 0.540i)T \)
5 \( 1 + (-0.415 - 0.909i)T \)
7 \( 1 + (0.654 - 0.755i)T \)
11 \( 1 + (-0.841 + 0.540i)T \)
13 \( 1 + (-0.654 - 0.755i)T \)
17 \( 1 - T \)
19 \( 1 + (0.415 - 0.909i)T \)
23 \( 1 + (-0.841 - 0.540i)T \)
29 \( 1 + (-0.841 - 0.540i)T \)
31 \( 1 + (0.142 - 0.989i)T \)
37 \( 1 + (0.654 - 0.755i)T \)
41 \( 1 + (0.142 - 0.989i)T \)
43 \( 1 + (-0.415 - 0.909i)T \)
47 \( 1 + (-0.142 - 0.989i)T \)
53 \( 1 + T \)
59 \( 1 + (0.415 - 0.909i)T \)
61 \( 1 + (0.959 - 0.281i)T \)
67 \( 1 + (-0.415 - 0.909i)T \)
71 \( 1 + (0.959 + 0.281i)T \)
73 \( 1 + (-0.142 - 0.989i)T \)
79 \( 1 + (0.415 + 0.909i)T \)
83 \( 1 + (0.142 + 0.989i)T \)
89 \( 1 + T \)
97 \( 1 + (-0.415 + 0.909i)T \)
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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

−20.05892046222915677179826293862, −19.503600104119131197514897882887, −18.72334112803602809573002644175, −18.33086344534614244403572868102, −17.6196396697039509136892236748, −16.304540655236124622530082462339, −16.04006357498356132276853112642, −15.18074574244235738352337591616, −14.6645176244031940405188783084, −14.19806639516198014481508228055, −13.28249275791135143619460487674, −11.81397473243164564503783138509, −11.2692382720395073021404843317, −10.48589192353655594050144400749, −9.811547231762627284004419180418, −9.04015951423182040405524839311, −8.21525669536504291643103943491, −7.809513847230660172384367507819, −7.00551731630878240314057097179, −5.99847863527935631399472580633, −5.1250920448300822091579027738, −4.254390048231938899347241676612, −3.01950635624281840215151908861, −2.36778378525778925066518824024, −1.564483715721872473105343600851, 0.39070433008403345601319797101, 0.59710610644646655651782430329, 2.088134349217820414353111046162, 2.276862161459138279966811696551, 3.67970112097163458766234426664, 4.281075146519240878897872540608, 5.2628871607439755950619098605, 6.81742923704934259214589844006, 7.55117478813505544991455735293, 7.92562481998612950983163001919, 8.61419838342193607635110737151, 9.43475516768282409421603860479, 10.10642894456023992463622953936, 11.0345468977330518621479123310, 11.8345249883202448275861252257, 12.608388962121837844767571554266, 13.18671685735659817046129873966, 13.73596764995295090856101203262, 15.12985555592808210532204294938, 15.4463867894088113936390632294, 16.464862159592218057747758345252, 17.340546168713184536832976574325, 17.811958670772973688584732325665, 18.453251047503815452891069860220, 19.471444030716835978853604393904

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