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Properties

Label 1-4009-4009.1067-r0-0-0
Degree $1$
Conductor $4009$
Sign $-0.930 - 0.365i$
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  + (0.921 − 0.388i)2-s + (−0.797 − 0.603i)3-s + (0.698 − 0.715i)4-s + (0.542 − 0.840i)5-s + (−0.969 − 0.246i)6-s + (0.733 + 0.680i)7-s + (0.365 − 0.930i)8-s + (0.270 + 0.962i)9-s + (0.173 − 0.984i)10-s + (−0.733 + 0.680i)11-s + (−0.988 + 0.149i)12-s + (−0.980 + 0.198i)13-s + (0.939 + 0.342i)14-s + (−0.939 + 0.342i)15-s + (−0.0249 − 0.999i)16-s + (0.661 − 0.749i)17-s + ⋯
L(s)  = 1  + (0.921 − 0.388i)2-s + (−0.797 − 0.603i)3-s + (0.698 − 0.715i)4-s + (0.542 − 0.840i)5-s + (−0.969 − 0.246i)6-s + (0.733 + 0.680i)7-s + (0.365 − 0.930i)8-s + (0.270 + 0.962i)9-s + (0.173 − 0.984i)10-s + (−0.733 + 0.680i)11-s + (−0.988 + 0.149i)12-s + (−0.980 + 0.198i)13-s + (0.939 + 0.342i)14-s + (−0.939 + 0.342i)15-s + (−0.0249 − 0.999i)16-s + (0.661 − 0.749i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4009 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.930 - 0.365i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4009 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.930 - 0.365i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4009\)    =    \(19 \cdot 211\)
Sign: $-0.930 - 0.365i$
Analytic conductor: \(18.6177\)
Root analytic conductor: \(18.6177\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4009} (1067, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4009,\ (0:\ ),\ -0.930 - 0.365i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4384614635 - 2.317059133i\)
\(L(\frac12)\) \(\approx\) \(0.4384614635 - 2.317059133i\)
\(L(1)\) \(\approx\) \(1.232474175 - 0.9304328259i\)
\(L(1)\) \(\approx\) \(1.232474175 - 0.9304328259i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad19 \( 1 \)
211 \( 1 \)
good2 \( 1 + (0.921 - 0.388i)T \)
3 \( 1 + (-0.797 - 0.603i)T \)
5 \( 1 + (0.542 - 0.840i)T \)
7 \( 1 + (0.733 + 0.680i)T \)
11 \( 1 + (-0.733 + 0.680i)T \)
13 \( 1 + (-0.980 + 0.198i)T \)
17 \( 1 + (0.661 - 0.749i)T \)
23 \( 1 + (0.939 - 0.342i)T \)
29 \( 1 + (-0.661 - 0.749i)T \)
31 \( 1 + (-0.988 + 0.149i)T \)
37 \( 1 + (0.222 - 0.974i)T \)
41 \( 1 + (0.980 + 0.198i)T \)
43 \( 1 + (-0.318 - 0.947i)T \)
47 \( 1 + (-0.583 + 0.811i)T \)
53 \( 1 + (-0.995 - 0.0995i)T \)
59 \( 1 + (0.661 - 0.749i)T \)
61 \( 1 + (0.939 - 0.342i)T \)
67 \( 1 + (0.921 + 0.388i)T \)
71 \( 1 + (0.939 + 0.342i)T \)
73 \( 1 + (-0.411 + 0.911i)T \)
79 \( 1 + (0.0249 + 0.999i)T \)
83 \( 1 + (-0.5 + 0.866i)T \)
89 \( 1 + (0.456 - 0.889i)T \)
97 \( 1 + (-0.583 - 0.811i)T \)
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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

−18.55695151629292129650114523118, −17.8247649682977541714868749740, −17.21321152180644035510114192178, −16.75215355928149055029398674754, −16.11321327779205558485013156256, −15.084194978271415238072282814574, −14.75415518566801704439180217744, −14.28273838314371687889330067687, −13.219678778050966768139938207297, −12.86460242468626597242383750106, −11.77798277042520778861513604827, −11.19864255441465048238286748520, −10.660254398018840681887233136983, −10.17381141463685513990379059913, −9.167658851040650644590119821305, −7.95518908696794023814597699033, −7.42317752691835881210399723165, −6.679571654888663789238520986894, −5.93893377415249368577272928924, −5.19723535228450408535924824074, −4.9152372941795089179241782891, −3.72112145585887430267649123966, −3.30435458285579230604259043118, −2.3039961849434981276802037230, −1.244331451212976048016115282281, 0.4996251302698225540716928080, 1.53663714519389747007746308299, 2.17062393562288984077323058430, 2.687616400047208596923969859067, 4.23608811157852121126094630763, 4.90540175783361759009856929859, 5.380584783168608218194351362091, 5.71623336793683461323822244920, 6.8793032550312964567550133720, 7.44269509320494062918256015765, 8.250482092424075941340956446688, 9.520092488332356132448650084623, 9.85127567744003009964152116628, 10.998052720800896815577318196009, 11.391589320291510258010004012076, 12.27687066928820598588104491940, 12.67936871811445852197999495101, 13.010238009373244356274099164999, 14.12011099098250475099475384394, 14.48164099139443033993660953800, 15.46950073886119461591815039213, 16.10412725149670089851220222423, 16.86137803149491938233231053943, 17.48384626121233157933585092313, 18.2224856203931067248540507152

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