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Properties

Label 1-2653-2653.1136-r1-0-0
Degree $1$
Conductor $2653$
Sign $-0.895 + 0.444i$
Analytic cond. $285.104$
Root an. cond. $285.104$
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  + (0.5 + 0.866i)2-s + (0.5 − 0.866i)3-s + (−0.5 + 0.866i)4-s + (−0.5 − 0.866i)5-s + 6-s − 8-s + (−0.5 − 0.866i)9-s + (0.5 − 0.866i)10-s + (0.5 − 0.866i)11-s + (0.5 + 0.866i)12-s − 13-s − 15-s + (−0.5 − 0.866i)16-s + (0.5 − 0.866i)17-s + (0.5 − 0.866i)18-s + (−0.5 − 0.866i)19-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)2-s + (0.5 − 0.866i)3-s + (−0.5 + 0.866i)4-s + (−0.5 − 0.866i)5-s + 6-s − 8-s + (−0.5 − 0.866i)9-s + (0.5 − 0.866i)10-s + (0.5 − 0.866i)11-s + (0.5 + 0.866i)12-s − 13-s − 15-s + (−0.5 − 0.866i)16-s + (0.5 − 0.866i)17-s + (0.5 − 0.866i)18-s + (−0.5 − 0.866i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2653 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.895 + 0.444i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2653 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.895 + 0.444i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2653\)    =    \(7 \cdot 379\)
Sign: $-0.895 + 0.444i$
Analytic conductor: \(285.104\)
Root analytic conductor: \(285.104\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2653} (1136, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2653,\ (1:\ ),\ -0.895 + 0.444i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.2894365087 - 1.235528582i\)
\(L(\frac12)\) \(\approx\) \(-0.2894365087 - 1.235528582i\)
\(L(1)\) \(\approx\) \(1.077489175 - 0.2983175472i\)
\(L(1)\) \(\approx\) \(1.077489175 - 0.2983175472i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
379 \( 1 \)
good2 \( 1 + (0.5 + 0.866i)T \)
3 \( 1 + (0.5 - 0.866i)T \)
5 \( 1 + (-0.5 - 0.866i)T \)
11 \( 1 + (0.5 - 0.866i)T \)
13 \( 1 - T \)
17 \( 1 + (0.5 - 0.866i)T \)
19 \( 1 + (-0.5 - 0.866i)T \)
23 \( 1 + (-0.5 - 0.866i)T \)
29 \( 1 - T \)
31 \( 1 + (0.5 - 0.866i)T \)
37 \( 1 + (-0.5 - 0.866i)T \)
41 \( 1 + T \)
43 \( 1 - T \)
47 \( 1 + (0.5 + 0.866i)T \)
53 \( 1 + (0.5 - 0.866i)T \)
59 \( 1 + (0.5 - 0.866i)T \)
61 \( 1 + (-0.5 - 0.866i)T \)
67 \( 1 + (-0.5 + 0.866i)T \)
71 \( 1 - T \)
73 \( 1 + (0.5 - 0.866i)T \)
79 \( 1 + (-0.5 - 0.866i)T \)
83 \( 1 + T \)
89 \( 1 + (0.5 + 0.866i)T \)
97 \( 1 + T \)
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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

−19.74884619532890708818012555596, −19.16185314172346021332314315572, −18.44348018769374877695337104997, −17.45665140900127933387334217689, −16.76123225999776325423995133068, −15.612109304568464272452673011884, −14.98839407284286690121478459384, −14.687625404646810107441425778120, −14.0653149238610053952009314110, −13.173767647540448710593614392977, −12.05058253561567895580370477295, −11.88912519793084273617401268824, −10.7378164113370750125958147141, −10.2378689658682696250400396055, −9.802194448791549531025317663579, −8.91362080514195446361016952228, −7.98133443078392432636703158692, −7.176616944852387979032039293806, −6.097006350360254706685558354784, −5.27151461796858486417537627055, −4.3170670644905237640216047379, −3.856465346695122580305326491120, −3.15107891524638554260006909583, −2.29306026091443629352588316158, −1.56288172012744771420297650147, 0.22420386904056536672222333324, 0.63962837752425527590496931811, 2.11184196799349573426181428471, 3.005562367302350729297128176568, 3.859142015362288827088251973698, 4.64691779428515980575511602965, 5.49989772996511209497782490589, 6.26829933081384671801192448718, 7.12137852011322353890180933974, 7.70239063944592213837667438398, 8.3493876698564513606807396986, 9.05990730300263621321045217582, 9.55437394622400320025300266325, 11.232824144474331345641991931904, 11.94089543215699220083335194151, 12.43372784218633542250419815679, 13.17401245282227175549899667558, 13.72021312221937215031675364106, 14.54591703179930507714266806272, 14.979973422299770348028719392621, 16.00733891288893964728043590967, 16.541694165066242468292976996880, 17.24998631987843060397234555877, 17.830532753953775718341274492517, 18.91655283484630373293530701655

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