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Properties

Label 1-4009-4009.1072-r0-0-0
Degree $1$
Conductor $4009$
Sign $-0.935 + 0.353i$
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  + (0.753 + 0.657i)2-s + (0.858 − 0.512i)3-s + (0.134 + 0.990i)4-s + (0.0149 − 0.999i)5-s + (0.983 + 0.178i)6-s + (−0.193 − 0.981i)7-s + (−0.550 + 0.834i)8-s + (0.473 − 0.880i)9-s + (0.669 − 0.743i)10-s + (−0.995 − 0.0896i)11-s + (0.623 + 0.781i)12-s + (−0.251 + 0.967i)13-s + (0.5 − 0.866i)14-s + (−0.5 − 0.866i)15-s + (−0.963 + 0.266i)16-s + (0.0448 + 0.998i)17-s + ⋯
L(s)  = 1  + (0.753 + 0.657i)2-s + (0.858 − 0.512i)3-s + (0.134 + 0.990i)4-s + (0.0149 − 0.999i)5-s + (0.983 + 0.178i)6-s + (−0.193 − 0.981i)7-s + (−0.550 + 0.834i)8-s + (0.473 − 0.880i)9-s + (0.669 − 0.743i)10-s + (−0.995 − 0.0896i)11-s + (0.623 + 0.781i)12-s + (−0.251 + 0.967i)13-s + (0.5 − 0.866i)14-s + (−0.5 − 0.866i)15-s + (−0.963 + 0.266i)16-s + (0.0448 + 0.998i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4009 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.935 + 0.353i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4009 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.935 + 0.353i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4009\)    =    \(19 \cdot 211\)
Sign: $-0.935 + 0.353i$
Analytic conductor: \(18.6177\)
Root analytic conductor: \(18.6177\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4009} (1072, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4009,\ (0:\ ),\ -0.935 + 0.353i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1209348614 + 0.6612934972i\)
\(L(\frac12)\) \(\approx\) \(0.1209348614 + 0.6612934972i\)
\(L(1)\) \(\approx\) \(1.444740110 + 0.1908380257i\)
\(L(1)\) \(\approx\) \(1.444740110 + 0.1908380257i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad19 \( 1 \)
211 \( 1 \)
good2 \( 1 + (0.753 + 0.657i)T \)
3 \( 1 + (0.858 - 0.512i)T \)
5 \( 1 + (0.0149 - 0.999i)T \)
7 \( 1 + (-0.193 - 0.981i)T \)
11 \( 1 + (-0.995 - 0.0896i)T \)
13 \( 1 + (-0.251 + 0.967i)T \)
17 \( 1 + (0.0448 + 0.998i)T \)
23 \( 1 + (0.104 + 0.994i)T \)
29 \( 1 + (-0.963 + 0.266i)T \)
31 \( 1 + (-0.988 + 0.149i)T \)
37 \( 1 + (-0.420 + 0.907i)T \)
41 \( 1 + (-0.163 + 0.986i)T \)
43 \( 1 + (-0.988 + 0.149i)T \)
47 \( 1 + (0.525 - 0.850i)T \)
53 \( 1 + (0.925 - 0.379i)T \)
59 \( 1 + (-0.936 + 0.351i)T \)
61 \( 1 + (-0.309 - 0.951i)T \)
67 \( 1 + (-0.733 - 0.680i)T \)
71 \( 1 + (-0.669 + 0.743i)T \)
73 \( 1 + (0.0747 - 0.997i)T \)
79 \( 1 + (0.447 - 0.894i)T \)
83 \( 1 + (0.669 + 0.743i)T \)
89 \( 1 + (-0.599 - 0.800i)T \)
97 \( 1 + (-0.280 + 0.959i)T \)
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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

−18.374202406727617154056122291629, −18.01579897590680321411315615132, −16.47066222489429759848286684635, −15.70425553794225573381048658099, −15.23205948763526003638552013732, −14.8437652192422336815239725804, −14.112488067767251541057644076807, −13.448562921203312077489437221629, −12.77024680021003128155525728020, −12.13398215356848003349660967199, −11.110431895657856504384899021961, −10.63973896390654393631729906185, −10.01181011853265448084341410270, −9.37291622279840125750955498494, −8.61018379663827760602593981502, −7.560700039604105495675695845663, −7.042045718865947919197694057298, −5.74963420803267980178780087514, −5.44950377309160635759187428845, −4.53272696528658967194765725593, −3.58502970829634459045981333778, −2.90524876930054206031373740936, −2.54124438829085199433950573777, −1.90615496367996279763928900671, −0.10526797884907243034127016849, 1.45639711700234208838381243641, 2.0380238111969073178781036207, 3.29434747513717822746307323256, 3.74628940261866557914212100467, 4.5475310326062448201970780202, 5.25389585268610625189714247868, 6.17720306585955276203791021240, 6.952307668875456122055498932854, 7.63982767626052844011357380639, 8.06561001414100151278245181065, 8.88389509517217101843180731592, 9.49233321595348275409687058545, 10.463523462694871751882557459523, 11.53592542924296546715252023032, 12.24951773708439867583835157468, 12.99448802645927743049234282420, 13.39352344479680967118697468537, 13.74662831516508893154880513131, 14.72799369309008610697142473414, 15.209407121977178271761776560285, 16.06254949017717705247346215813, 16.698966291214738362758943096253, 17.13366489306726355635797939966, 18.007760344252359663386549204540, 18.7785028722704273671336627850

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