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Properties

Label 1-4009-4009.1077-r0-0-0
Degree $1$
Conductor $4009$
Sign $0.823 + 0.567i$
Analytic cond. $18.6177$
Root an. cond. $18.6177$
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  + (0.943 + 0.332i)2-s + (0.996 − 0.0797i)3-s + (0.778 + 0.627i)4-s + (0.814 − 0.579i)5-s + (0.966 + 0.256i)6-s + (−0.936 − 0.351i)7-s + (0.525 + 0.850i)8-s + (0.987 − 0.158i)9-s + (0.961 − 0.275i)10-s + (0.887 + 0.460i)11-s + (0.826 + 0.563i)12-s + (0.534 + 0.845i)13-s + (−0.766 − 0.642i)14-s + (0.766 − 0.642i)15-s + (0.212 + 0.977i)16-s + (0.0647 + 0.997i)17-s + ⋯
L(s)  = 1  + (0.943 + 0.332i)2-s + (0.996 − 0.0797i)3-s + (0.778 + 0.627i)4-s + (0.814 − 0.579i)5-s + (0.966 + 0.256i)6-s + (−0.936 − 0.351i)7-s + (0.525 + 0.850i)8-s + (0.987 − 0.158i)9-s + (0.961 − 0.275i)10-s + (0.887 + 0.460i)11-s + (0.826 + 0.563i)12-s + (0.534 + 0.845i)13-s + (−0.766 − 0.642i)14-s + (0.766 − 0.642i)15-s + (0.212 + 0.977i)16-s + (0.0647 + 0.997i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4009 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.823 + 0.567i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4009 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.823 + 0.567i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4009\)    =    \(19 \cdot 211\)
Sign: $0.823 + 0.567i$
Analytic conductor: \(18.6177\)
Root analytic conductor: \(18.6177\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4009} (1077, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4009,\ (0:\ ),\ 0.823 + 0.567i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(5.788963290 + 1.801631443i\)
\(L(\frac12)\) \(\approx\) \(5.788963290 + 1.801631443i\)
\(L(1)\) \(\approx\) \(2.964608831 + 0.5570311216i\)
\(L(1)\) \(\approx\) \(2.964608831 + 0.5570311216i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad19 \( 1 \)
211 \( 1 \)
good2 \( 1 + (0.943 + 0.332i)T \)
3 \( 1 + (0.996 - 0.0797i)T \)
5 \( 1 + (0.814 - 0.579i)T \)
7 \( 1 + (-0.936 - 0.351i)T \)
11 \( 1 + (0.887 + 0.460i)T \)
13 \( 1 + (0.534 + 0.845i)T \)
17 \( 1 + (0.0647 + 0.997i)T \)
23 \( 1 + (0.615 - 0.788i)T \)
29 \( 1 + (-0.534 - 0.845i)T \)
31 \( 1 + (-0.900 + 0.433i)T \)
37 \( 1 + (0.842 + 0.538i)T \)
41 \( 1 + (0.346 - 0.937i)T \)
43 \( 1 + (-0.969 - 0.246i)T \)
47 \( 1 + (-0.982 + 0.188i)T \)
53 \( 1 + (-0.999 - 0.0199i)T \)
59 \( 1 + (-0.638 - 0.769i)T \)
61 \( 1 + (0.719 + 0.694i)T \)
67 \( 1 + (0.878 + 0.478i)T \)
71 \( 1 + (-0.559 + 0.829i)T \)
73 \( 1 + (0.921 + 0.388i)T \)
79 \( 1 + (0.203 - 0.979i)T \)
83 \( 1 + (-0.809 - 0.587i)T \)
89 \( 1 + (0.114 + 0.993i)T \)
97 \( 1 + (-0.973 + 0.227i)T \)
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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

−18.48902901532867116896851468807, −18.15152664959175472642386124919, −16.72365234980062701111874884962, −16.239001850080285952061614170321, −15.34652464213975870253314452782, −14.924836698984216714383985872092, −14.182548530691753378043080741970, −13.70328676829089416333598560693, −12.93909033535046068840383512821, −12.75309502712575407385706374808, −11.435788765676622287461569242480, −10.98191189015051480979651330269, −9.976857021576612617375777401433, −9.51257228477012159464609913603, −9.014984684625154333051794885780, −7.77220161533734561730696518719, −6.97353236268539897648967161593, −6.40183715847674130693368390451, −5.67873499710633317194508442863, −4.93498183318822529209444632546, −3.70042409875615803978332315161, −3.27184886904509628900335183283, −2.81588236889329897165638083121, −1.893977389226467939659870061704, −1.09552648438492429668329784795, 1.329338042996151078383755323632, 1.90799266572951872410193396400, 2.75128328694428438610976763114, 3.7323882403824834502343835439, 4.08252639458808181284758465257, 4.91232203198607867188538418801, 6.03377125653949871573073078305, 6.56041610422045526476095159621, 7.04915595502722469039530268989, 8.106090081146665924828108074, 8.79844239605583980225782102578, 9.432904668511688057396230074011, 10.0964564904245227100446194215, 11.01919599244457819206741944047, 12.05834394885246587037388502293, 12.874517499933090265121685268832, 12.99754715359086262128309644980, 13.80606686813465953932589053898, 14.37574058977457366022946738951, 14.90260109987229520065299406063, 15.75982450645453790224448266480, 16.47804645149711353088268393347, 16.89800632066159129921656549602, 17.642571554250074644071850943, 18.78749722140033257748526543912

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