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Properties

Label 1-4009-4009.1002-r0-0-0
Degree $1$
Conductor $4009$
Sign $0.369 + 0.929i$
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  + (0.726 − 0.687i)2-s + (0.607 + 0.794i)3-s + (0.0548 − 0.998i)4-s + (0.976 − 0.217i)5-s + (0.987 + 0.158i)6-s + (−0.998 − 0.0598i)7-s + (−0.646 − 0.762i)8-s + (−0.261 + 0.965i)9-s + (0.559 − 0.829i)10-s + (0.712 + 0.701i)11-s + (0.826 − 0.563i)12-s + (−0.896 − 0.442i)13-s + (−0.766 + 0.642i)14-s + (0.766 + 0.642i)15-s + (−0.993 − 0.109i)16-s + (−0.999 + 0.0398i)17-s + ⋯
L(s)  = 1  + (0.726 − 0.687i)2-s + (0.607 + 0.794i)3-s + (0.0548 − 0.998i)4-s + (0.976 − 0.217i)5-s + (0.987 + 0.158i)6-s + (−0.998 − 0.0598i)7-s + (−0.646 − 0.762i)8-s + (−0.261 + 0.965i)9-s + (0.559 − 0.829i)10-s + (0.712 + 0.701i)11-s + (0.826 − 0.563i)12-s + (−0.896 − 0.442i)13-s + (−0.766 + 0.642i)14-s + (0.766 + 0.642i)15-s + (−0.993 − 0.109i)16-s + (−0.999 + 0.0398i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4009 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.369 + 0.929i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4009 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.369 + 0.929i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4009\)    =    \(19 \cdot 211\)
Sign: $0.369 + 0.929i$
Analytic conductor: \(18.6177\)
Root analytic conductor: \(18.6177\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4009} (1002, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4009,\ (0:\ ),\ 0.369 + 0.929i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.874843429 + 1.271776843i\)
\(L(\frac12)\) \(\approx\) \(1.874843429 + 1.271776843i\)
\(L(1)\) \(\approx\) \(1.668462106 - 0.07969285147i\)
\(L(1)\) \(\approx\) \(1.668462106 - 0.07969285147i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad19 \( 1 \)
211 \( 1 \)
good2 \( 1 + (0.726 - 0.687i)T \)
3 \( 1 + (0.607 + 0.794i)T \)
5 \( 1 + (0.976 - 0.217i)T \)
7 \( 1 + (-0.998 - 0.0598i)T \)
11 \( 1 + (0.712 + 0.701i)T \)
13 \( 1 + (-0.896 - 0.442i)T \)
17 \( 1 + (-0.999 + 0.0398i)T \)
23 \( 1 + (0.882 + 0.469i)T \)
29 \( 1 + (-0.831 + 0.555i)T \)
31 \( 1 + (0.0747 + 0.997i)T \)
37 \( 1 + (-0.712 + 0.701i)T \)
41 \( 1 + (-0.784 + 0.619i)T \)
43 \( 1 + (-0.969 + 0.246i)T \)
47 \( 1 + (0.905 - 0.424i)T \)
53 \( 1 + (-0.289 + 0.957i)T \)
59 \( 1 + (0.784 - 0.619i)T \)
61 \( 1 + (-0.559 + 0.829i)T \)
67 \( 1 + (-0.0249 + 0.999i)T \)
71 \( 1 + (0.719 + 0.694i)T \)
73 \( 1 + (0.921 - 0.388i)T \)
79 \( 1 + (0.863 + 0.504i)T \)
83 \( 1 + (-0.104 + 0.994i)T \)
89 \( 1 + (-0.999 - 0.00997i)T \)
97 \( 1 + (0.999 + 0.0199i)T \)
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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

−18.43311420313653581155762784925, −17.346408682020277961247507144739, −17.09467520795801842148448291665, −16.40811254601047376233410891903, −15.32396064423748399180740634037, −14.894281652511642821180935414487, −14.06794875703105961073454088809, −13.64651059168024484671209956834, −13.11230135828597512470472154962, −12.50995481294780040510085256016, −11.78794394074523065190313599038, −10.9356446852530967209463385278, −9.70134386886774620422001815125, −9.114904268466615751741087826603, −8.702653705694928163942235112366, −7.586242668366725061408285001896, −6.81832185098731461002922638683, −6.54222846004400818255121568345, −5.88520374148154129491135044729, −5.0263247679333113547150200927, −3.894101384179445637787128944757, −3.28407661485167925905701631816, −2.41861630913257663503812639809, −1.97216362501043948222541172838, −0.387096518482494618706116502747, 1.30780012372968177673086696087, 2.13048606879319476075335585395, 2.83020825746793922117111638397, 3.44930135886767785316188316377, 4.3222597366824170015922716216, 5.04670726643747759082410489318, 5.52662013012067974616587091380, 6.67935620846345802226472589122, 7.024689921705708004954524225387, 8.58405533446432835673487113164, 9.28821426014842440286761799048, 9.638708810358802751190773970047, 10.266718265291795222310432981480, 10.82702444504050065490360787361, 11.85188761689722122196704647269, 12.6690345402693901757329362025, 13.15942567945024634968060689269, 13.77585112769833902284846214902, 14.42509652011907686843532526678, 15.202591509208764936454269883249, 15.48540088419226874623134983068, 16.61362535783055122479961541399, 17.0796331021405695968546224249, 17.98051213793329133028930575961, 18.900208492241951970623041387290

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