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Properties

Label 1-1009-1009.15-r0-0-0
Degree $1$
Conductor $1009$
Sign $-0.104 - 0.994i$
Analytic cond. $4.68577$
Root an. cond. $4.68577$
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  + (−0.542 − 0.840i)2-s + (−0.955 − 0.294i)3-s + (−0.411 + 0.911i)4-s + (0.969 + 0.246i)5-s + (0.270 + 0.962i)6-s + (−0.124 + 0.992i)7-s + (0.988 − 0.149i)8-s + (0.826 + 0.563i)9-s + (−0.318 − 0.947i)10-s + (0.246 − 0.969i)11-s + (0.661 − 0.749i)12-s + (0.866 − 0.5i)13-s + (0.900 − 0.433i)14-s + (−0.853 − 0.521i)15-s + (−0.661 − 0.749i)16-s + (0.749 − 0.661i)17-s + ⋯
L(s)  = 1  + (−0.542 − 0.840i)2-s + (−0.955 − 0.294i)3-s + (−0.411 + 0.911i)4-s + (0.969 + 0.246i)5-s + (0.270 + 0.962i)6-s + (−0.124 + 0.992i)7-s + (0.988 − 0.149i)8-s + (0.826 + 0.563i)9-s + (−0.318 − 0.947i)10-s + (0.246 − 0.969i)11-s + (0.661 − 0.749i)12-s + (0.866 − 0.5i)13-s + (0.900 − 0.433i)14-s + (−0.853 − 0.521i)15-s + (−0.661 − 0.749i)16-s + (0.749 − 0.661i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1009 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.104 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1009 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.104 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1009\)
Sign: $-0.104 - 0.994i$
Analytic conductor: \(4.68577\)
Root analytic conductor: \(4.68577\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1009} (15, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1009,\ (0:\ ),\ -0.104 - 0.994i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6099960359 - 0.6777143092i\)
\(L(\frac12)\) \(\approx\) \(0.6099960359 - 0.6777143092i\)
\(L(1)\) \(\approx\) \(0.6692252725 - 0.3160716385i\)
\(L(1)\) \(\approx\) \(0.6692252725 - 0.3160716385i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad1009 \( 1 \)
good2 \( 1 + (-0.542 - 0.840i)T \)
3 \( 1 + (-0.955 - 0.294i)T \)
5 \( 1 + (0.969 + 0.246i)T \)
7 \( 1 + (-0.124 + 0.992i)T \)
11 \( 1 + (0.246 - 0.969i)T \)
13 \( 1 + (0.866 - 0.5i)T \)
17 \( 1 + (0.749 - 0.661i)T \)
19 \( 1 + (-0.866 - 0.5i)T \)
23 \( 1 + (-0.930 + 0.365i)T \)
29 \( 1 + (0.797 - 0.603i)T \)
31 \( 1 + (-0.840 - 0.542i)T \)
37 \( 1 + (0.939 + 0.342i)T \)
41 \( 1 + (-0.939 - 0.342i)T \)
43 \( 1 + (0.866 - 0.5i)T \)
47 \( 1 + (-0.680 - 0.733i)T \)
53 \( 1 + (0.889 + 0.456i)T \)
59 \( 1 + (-0.433 + 0.900i)T \)
61 \( 1 + (-0.294 - 0.955i)T \)
67 \( 1 + (-0.124 - 0.992i)T \)
71 \( 1 + (-0.542 - 0.840i)T \)
73 \( 1 + (0.997 - 0.0747i)T \)
79 \( 1 + (0.521 - 0.853i)T \)
83 \( 1 + (0.947 + 0.318i)T \)
89 \( 1 + (-0.962 + 0.270i)T \)
97 \( 1 + (0.0995 - 0.995i)T \)
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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

−21.96936797724889249420071793885, −21.10551818176442379474474701924, −20.3209959408855173404518854749, −19.34838353307418932333574922856, −18.16269952143431461505986577556, −17.89172227005038775737874147245, −16.94868633723644373801939052023, −16.61088413072500547282500788008, −15.94184431983786825473939374325, −14.701174235615619876681212745452, −14.21169113309883825947572525300, −13.11557895710001707004626515071, −12.46553237262594633673196835512, −11.080493459256944403317069174826, −10.27674924345598524951246309297, −9.96543272297322829176620994011, −9.02128972814479407397462397655, −7.95906192498599396945410000228, −6.82486279319527384724743173088, −6.37065096922999572404244907161, −5.597151011974843236476944480109, −4.571985849901226163478464221562, −3.9538202035651055928439049679, −1.7421601007609026012238869071, −1.12017935276564512833230609224, 0.63348863296840241905981188398, 1.73899680411672186604911064581, 2.58513708243490122111356120474, 3.624051566670485567038730515455, 4.99390377406052157278843823840, 5.87380983520022701856859785402, 6.41873861221961476129715191544, 7.72536465155083138699224221023, 8.656779203521336679882563866263, 9.47050682903250482173402111426, 10.31095834382020952967414888709, 11.012762167897529557238351663138, 11.74195130630894541029896447786, 12.4602638400904328308497310852, 13.36615234014635979109878736584, 13.82814673003157048379654061662, 15.298778064216246768700840087122, 16.327389280463283525149356307449, 16.87416088752964069341452038298, 17.806329904839664967491382978231, 18.357168144531722376126945877665, 18.75646908558945032121702288832, 19.67713596835758021380228004783, 20.922899918516025477383875958792, 21.489217382697317262247174247318

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