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Properties

Label 1-1005-1005.584-r0-0-0
Degree $1$
Conductor $1005$
Sign $-0.720 - 0.693i$
Analytic cond. $4.66720$
Root an. cond. $4.66720$
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  + (−0.580 − 0.814i)2-s + (−0.327 + 0.945i)4-s + (−0.995 + 0.0950i)7-s + (0.959 − 0.281i)8-s + (0.928 − 0.371i)11-s + (0.723 − 0.690i)13-s + (0.654 + 0.755i)14-s + (−0.786 − 0.618i)16-s + (−0.327 − 0.945i)17-s + (−0.995 − 0.0950i)19-s + (−0.841 − 0.540i)22-s + (0.0475 + 0.998i)23-s + (−0.981 − 0.189i)26-s + (0.235 − 0.971i)28-s + (0.5 + 0.866i)29-s + ⋯
L(s)  = 1  + (−0.580 − 0.814i)2-s + (−0.327 + 0.945i)4-s + (−0.995 + 0.0950i)7-s + (0.959 − 0.281i)8-s + (0.928 − 0.371i)11-s + (0.723 − 0.690i)13-s + (0.654 + 0.755i)14-s + (−0.786 − 0.618i)16-s + (−0.327 − 0.945i)17-s + (−0.995 − 0.0950i)19-s + (−0.841 − 0.540i)22-s + (0.0475 + 0.998i)23-s + (−0.981 − 0.189i)26-s + (0.235 − 0.971i)28-s + (0.5 + 0.866i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1005 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.720 - 0.693i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1005 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.720 - 0.693i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1005\)    =    \(3 \cdot 5 \cdot 67\)
Sign: $-0.720 - 0.693i$
Analytic conductor: \(4.66720\)
Root analytic conductor: \(4.66720\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1005} (584, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1005,\ (0:\ ),\ -0.720 - 0.693i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2668760292 - 0.6619728865i\)
\(L(\frac12)\) \(\approx\) \(0.2668760292 - 0.6619728865i\)
\(L(1)\) \(\approx\) \(0.6173956007 - 0.3188171324i\)
\(L(1)\) \(\approx\) \(0.6173956007 - 0.3188171324i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
67 \( 1 \)
good2 \( 1 + (-0.580 - 0.814i)T \)
7 \( 1 + (-0.995 + 0.0950i)T \)
11 \( 1 + (0.928 - 0.371i)T \)
13 \( 1 + (0.723 - 0.690i)T \)
17 \( 1 + (-0.327 - 0.945i)T \)
19 \( 1 + (-0.995 - 0.0950i)T \)
23 \( 1 + (0.0475 + 0.998i)T \)
29 \( 1 + (0.5 + 0.866i)T \)
31 \( 1 + (-0.723 - 0.690i)T \)
37 \( 1 + (0.5 - 0.866i)T \)
41 \( 1 + (0.981 + 0.189i)T \)
43 \( 1 + (-0.654 + 0.755i)T \)
47 \( 1 + (-0.888 - 0.458i)T \)
53 \( 1 + (0.654 + 0.755i)T \)
59 \( 1 + (0.959 - 0.281i)T \)
61 \( 1 + (-0.928 - 0.371i)T \)
71 \( 1 + (0.327 - 0.945i)T \)
73 \( 1 + (-0.928 - 0.371i)T \)
79 \( 1 + (-0.235 - 0.971i)T \)
83 \( 1 + (-0.786 - 0.618i)T \)
89 \( 1 + (-0.841 - 0.540i)T \)
97 \( 1 + (-0.5 + 0.866i)T \)
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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

−22.18014043678913813790335317683, −21.182270521261814898374780302266, −20.014229511614602773528717407021, −19.43496017891870303615721589704, −18.84966093612846886396349841459, −17.95581081098403386157607548811, −17.0073289892078249069882334711, −16.59976248103798920615759287709, −15.76179144331726447556700210736, −14.94230295199268458241085503053, −14.258193754016846437091256540597, −13.30760926085357778495507504217, −12.56631800168637215607162422816, −11.37565428783662516645834264593, −10.43978771523254350017763381096, −9.75792517856553792419166535565, −8.829226897015493150040935519475, −8.36459304798339660422558887863, −6.96953388784129323268390992725, −6.51853355457190039676498547066, −5.88927192034912474227229422356, −4.44318846524902375354539055079, −3.852122612194641291808934494223, −2.25212704040938051345169547555, −1.15527341236430814271972134339, 0.4229841684346037486256153325, 1.5966095901617205912535127423, 2.83138238249875088625514196663, 3.497313370360694709174349703932, 4.37839438440300121996406013259, 5.764657710677126077979476929, 6.69260544465436532187907741730, 7.59615092390251302655604410792, 8.72437527910132217493263005300, 9.2219703180323788229041799598, 10.02819982150143306964578958031, 10.98837042049881994445536558114, 11.565190960507346098950122371186, 12.61302387778235149947735222945, 13.14986286488932497736087475872, 13.946747852824538582914542506040, 15.13970345962078635613711111642, 16.19866966431235835245752829585, 16.59230947294744452422744633924, 17.67289047028197966020131671570, 18.25994684412980024720926646375, 19.185933065710661713576755942319, 19.74327558037989204249508669023, 20.318793404426102423113538207364, 21.410443717982105869503284114

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