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Properties

Label 1-4003-4003.913-r0-0-0
Degree $1$
Conductor $4003$
Sign $-0.331 - 0.943i$
Analytic cond. $18.5898$
Root an. cond. $18.5898$
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  + (−0.619 + 0.785i)2-s + (0.126 + 0.992i)3-s + (−0.232 − 0.972i)4-s + (0.590 + 0.806i)5-s + (−0.856 − 0.515i)6-s + (−0.773 + 0.633i)7-s + (0.907 + 0.419i)8-s + (−0.968 + 0.250i)9-s + (−0.999 − 0.0361i)10-s + (0.0541 − 0.998i)11-s + (0.935 − 0.353i)12-s + (0.647 + 0.762i)13-s + (−0.0180 − 0.999i)14-s + (−0.725 + 0.687i)15-s + (−0.891 + 0.452i)16-s + (−0.561 + 0.827i)17-s + ⋯
L(s)  = 1  + (−0.619 + 0.785i)2-s + (0.126 + 0.992i)3-s + (−0.232 − 0.972i)4-s + (0.590 + 0.806i)5-s + (−0.856 − 0.515i)6-s + (−0.773 + 0.633i)7-s + (0.907 + 0.419i)8-s + (−0.968 + 0.250i)9-s + (−0.999 − 0.0361i)10-s + (0.0541 − 0.998i)11-s + (0.935 − 0.353i)12-s + (0.647 + 0.762i)13-s + (−0.0180 − 0.999i)14-s + (−0.725 + 0.687i)15-s + (−0.891 + 0.452i)16-s + (−0.561 + 0.827i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4003 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.331 - 0.943i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4003 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.331 - 0.943i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4003\)
Sign: $-0.331 - 0.943i$
Analytic conductor: \(18.5898\)
Root analytic conductor: \(18.5898\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4003} (913, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4003,\ (0:\ ),\ -0.331 - 0.943i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.5432641876 + 0.7665196483i\)
\(L(\frac12)\) \(\approx\) \(-0.5432641876 + 0.7665196483i\)
\(L(1)\) \(\approx\) \(0.4238918112 + 0.6559942286i\)
\(L(1)\) \(\approx\) \(0.4238918112 + 0.6559942286i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad4003 \( 1 \)
good2 \( 1 + (-0.619 + 0.785i)T \)
3 \( 1 + (0.126 + 0.992i)T \)
5 \( 1 + (0.590 + 0.806i)T \)
7 \( 1 + (-0.773 + 0.633i)T \)
11 \( 1 + (0.0541 - 0.998i)T \)
13 \( 1 + (0.647 + 0.762i)T \)
17 \( 1 + (-0.561 + 0.827i)T \)
19 \( 1 + (0.997 + 0.0721i)T \)
23 \( 1 + (-0.891 + 0.452i)T \)
29 \( 1 + (0.997 - 0.0721i)T \)
31 \( 1 + (0.958 + 0.284i)T \)
37 \( 1 + (0.997 - 0.0721i)T \)
41 \( 1 + (0.0541 + 0.998i)T \)
43 \( 1 + (-0.922 + 0.386i)T \)
47 \( 1 + (-0.891 - 0.452i)T \)
53 \( 1 + (-0.891 - 0.452i)T \)
59 \( 1 + (-0.817 + 0.576i)T \)
61 \( 1 + (0.874 + 0.484i)T \)
67 \( 1 + (-0.0180 + 0.999i)T \)
71 \( 1 + (0.126 + 0.992i)T \)
73 \( 1 + (0.976 + 0.214i)T \)
79 \( 1 + (0.874 - 0.484i)T \)
83 \( 1 + (-0.619 + 0.785i)T \)
89 \( 1 + (-0.0180 - 0.999i)T \)
97 \( 1 + (-0.0180 + 0.999i)T \)
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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

−18.02725219298343989202961442197, −17.57616492118207897317359465259, −16.95804279861499823936234659352, −16.163764299132982667139205687719, −15.58340256757032198423487876330, −14.06510121199140462878394289096, −13.66415603954370388493976634159, −13.13687717080168201721101107001, −12.416459957455619542159351379834, −12.10232437475112373052550624038, −11.147409836361617733751387351643, −10.259628104182221239051832928862, −9.62045532127669713822102778063, −9.166892176274019042976367494019, −8.13023711851391584904197184395, −7.81268536705278453566897021583, −6.78153157836809740651732496546, −6.317732544155209634073130465449, −5.12664646346971284617535033918, −4.3578062486799080365324728756, −3.31486047578560689950617496608, −2.6236463243134335692006542931, −1.82718001023050854896682983392, −1.00809140469239925422416033120, −0.37135023824678097679196197776, 1.286777409958384294316072921548, 2.39650370320351275146974647016, 3.172055932018599164758007544459, 3.9529316743375250547815357107, 4.98183267846586443810161008328, 5.87986703254963808291792961462, 6.20264139888875541139846194601, 6.77182377784281301621216796445, 8.12568653777340201824966110662, 8.505146462648803843314103677538, 9.38951808966914301076742381580, 9.80483841019459457832323083412, 10.373223221145941153159161645112, 11.270379598817523669797411602418, 11.64263935735101636098449574007, 13.23757815473924517894864093166, 13.768098541712557091272560265513, 14.327399039186933255904470056455, 15.04462717731974668448216232089, 15.71771842933077965514841520894, 16.18867333651235407315726389082, 16.681117990995156225847320806354, 17.637378723327355424833296284669, 18.15009152894270336257953461266, 18.88084195946576201958424814196

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