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Properties

Label 1-1840-1840.1323-r0-0-0
Degree $1$
Conductor $1840$
Sign $0.955 + 0.294i$
Analytic cond. $8.54492$
Root an. cond. $8.54492$
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  + (0.959 + 0.281i)3-s + (0.989 + 0.142i)7-s + (0.841 + 0.540i)9-s + (0.909 − 0.415i)11-s + (−0.142 − 0.989i)13-s + (0.755 + 0.654i)17-s + (−0.755 + 0.654i)19-s + (0.909 + 0.415i)21-s + (0.654 + 0.755i)27-s + (−0.755 − 0.654i)29-s + (0.959 − 0.281i)31-s + (0.989 − 0.142i)33-s + (0.841 + 0.540i)37-s + (0.142 − 0.989i)39-s + (−0.841 + 0.540i)41-s + ⋯
L(s)  = 1  + (0.959 + 0.281i)3-s + (0.989 + 0.142i)7-s + (0.841 + 0.540i)9-s + (0.909 − 0.415i)11-s + (−0.142 − 0.989i)13-s + (0.755 + 0.654i)17-s + (−0.755 + 0.654i)19-s + (0.909 + 0.415i)21-s + (0.654 + 0.755i)27-s + (−0.755 − 0.654i)29-s + (0.959 − 0.281i)31-s + (0.989 − 0.142i)33-s + (0.841 + 0.540i)37-s + (0.142 − 0.989i)39-s + (−0.841 + 0.540i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.955 + 0.294i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.955 + 0.294i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1840\)    =    \(2^{4} \cdot 5 \cdot 23\)
Sign: $0.955 + 0.294i$
Analytic conductor: \(8.54492\)
Root analytic conductor: \(8.54492\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1840} (1323, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1840,\ (0:\ ),\ 0.955 + 0.294i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.921768054 + 0.4392502959i\)
\(L(\frac12)\) \(\approx\) \(2.921768054 + 0.4392502959i\)
\(L(1)\) \(\approx\) \(1.760334070 + 0.1656522490i\)
\(L(1)\) \(\approx\) \(1.760334070 + 0.1656522490i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
23 \( 1 \)
good3 \( 1 + (0.959 + 0.281i)T \)
7 \( 1 + (0.989 + 0.142i)T \)
11 \( 1 + (0.909 - 0.415i)T \)
13 \( 1 + (-0.142 - 0.989i)T \)
17 \( 1 + (0.755 + 0.654i)T \)
19 \( 1 + (-0.755 + 0.654i)T \)
29 \( 1 + (-0.755 - 0.654i)T \)
31 \( 1 + (0.959 - 0.281i)T \)
37 \( 1 + (0.841 + 0.540i)T \)
41 \( 1 + (-0.841 + 0.540i)T \)
43 \( 1 + (-0.959 - 0.281i)T \)
47 \( 1 - iT \)
53 \( 1 + (0.142 - 0.989i)T \)
59 \( 1 + (0.989 - 0.142i)T \)
61 \( 1 + (0.281 + 0.959i)T \)
67 \( 1 + (0.415 - 0.909i)T \)
71 \( 1 + (0.415 - 0.909i)T \)
73 \( 1 + (-0.755 + 0.654i)T \)
79 \( 1 + (-0.142 - 0.989i)T \)
83 \( 1 + (-0.841 - 0.540i)T \)
89 \( 1 + (-0.959 - 0.281i)T \)
97 \( 1 + (-0.540 - 0.841i)T \)
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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

−20.13580778140325189924588009016, −19.42119746542955802391707886207, −18.70472338305149579832467806879, −18.05755153682957523233937187224, −17.17657810408263238480827621673, −16.555075271891473166511607977015, −15.42776580797578142135639460444, −14.76478808814936396888552248255, −14.24512830288932894921096103370, −13.68225288216672884887249798596, −12.7210649273044846424213126125, −11.88311942725402426480065586331, −11.336019370108552543525725944293, −10.181502271871929852272767258524, −9.415882676232586683289895891259, −8.74004738952044985787153296066, −8.07145235358184146450325396618, −7.0431153190103130211971243088, −6.7845583547036100834131256673, −5.34241114346057018290195998019, −4.40131292361797283860814576132, −3.855706945273047865848982521882, −2.67418041770457206561305163370, −1.85446475997169841387796796051, −1.13062012620892667747048130715, 1.17865928206565280562312290604, 1.94453097566373084639320742543, 2.97784902217042800250674077108, 3.80565531127162240066287429117, 4.51878916201201195732256829073, 5.505828584381991865452568063030, 6.37909100563616498481971210555, 7.563169564274169351126566794289, 8.236216109865250365301646293937, 8.55749332785168898428134357590, 9.73116155606994746888104768212, 10.24015053019958016472406152000, 11.19017333957484908647877608155, 11.96912737609568892499119816421, 12.89757219593515162100092945201, 13.606610158741061437587114946103, 14.54927070074192524484039157860, 14.82183836323441711074415635421, 15.475307815456905067314062675982, 16.63164344568437285531872797661, 17.11864752506854034909958016018, 18.06888484773974463112090004523, 18.91682412498851014971113882035, 19.4038037377830389566434693436, 20.31199810808272479737772666843

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