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Properties

Label 1-1840-1840.1293-r0-0-0
Degree $1$
Conductor $1840$
Sign $-0.840 + 0.541i$
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  + (0.142 + 0.989i)3-s + (0.755 + 0.654i)7-s + (−0.959 + 0.281i)9-s + (0.540 + 0.841i)11-s + (0.654 + 0.755i)13-s + (0.909 + 0.415i)17-s + (−0.909 + 0.415i)19-s + (−0.540 + 0.841i)21-s + (−0.415 − 0.909i)27-s + (−0.909 − 0.415i)29-s + (−0.142 + 0.989i)31-s + (−0.755 + 0.654i)33-s + (−0.959 + 0.281i)37-s + (−0.654 + 0.755i)39-s + (0.959 + 0.281i)41-s + ⋯
L(s)  = 1  + (0.142 + 0.989i)3-s + (0.755 + 0.654i)7-s + (−0.959 + 0.281i)9-s + (0.540 + 0.841i)11-s + (0.654 + 0.755i)13-s + (0.909 + 0.415i)17-s + (−0.909 + 0.415i)19-s + (−0.540 + 0.841i)21-s + (−0.415 − 0.909i)27-s + (−0.909 − 0.415i)29-s + (−0.142 + 0.989i)31-s + (−0.755 + 0.654i)33-s + (−0.959 + 0.281i)37-s + (−0.654 + 0.755i)39-s + (0.959 + 0.281i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.840 + 0.541i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.840 + 0.541i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1840\)    =    \(2^{4} \cdot 5 \cdot 23\)
Sign: $-0.840 + 0.541i$
Analytic conductor: \(8.54492\)
Root analytic conductor: \(8.54492\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1840} (1293, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1840,\ (0:\ ),\ -0.840 + 0.541i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4965844034 + 1.689142190i\)
\(L(\frac12)\) \(\approx\) \(0.4965844034 + 1.689142190i\)
\(L(1)\) \(\approx\) \(0.9983756235 + 0.6970371942i\)
\(L(1)\) \(\approx\) \(0.9983756235 + 0.6970371942i\)

Euler product

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

−19.75498446707127363586591122227, −19.04355461675507908323948313243, −18.46633416078370350008481985366, −17.62229038224323066604311379267, −17.09002453133933464787488509605, −16.38655995278573636937437110249, −15.25954369490309717634804831941, −14.42028978604047653801176828001, −13.95613349034066451756209469457, −13.185733690262370321307419427196, −12.54515249756592300696370662549, −11.50537287283986148960195725878, −11.104731138829373252841195499045, −10.227878557030735064792589274733, −8.93126959963807006026820591359, −8.46838060339379615287891644161, −7.57028695914342214473113398971, −7.08109258286193678001634846645, −5.93841505102724377391843942252, −5.50387648106966017707827292851, −4.143585120603157599798063041999, −3.388705975658725281733335556953, −2.36739054272151774230423862373, −1.32686354101113301507032816724, −0.63622198851975273569076435643, 1.53400240141274911644020999839, 2.22953949184498942690613024100, 3.48987195617475834212553304957, 4.138622555495522612443560279409, 4.93207137075769432036659528309, 5.71784869125613679506472964003, 6.55997443702007789118390420383, 7.738023990627678955376450362113, 8.548830382387518604196956144907, 9.086499846194331045175656590667, 9.93054075635857530973483810744, 10.658414063017774321042291935343, 11.5013307308974340069445722206, 12.04120918593154601660075508153, 12.99198603497011778263539604748, 14.181202650792795053123582689169, 14.62726903259688492355277140899, 15.14528935445074877364631217296, 16.02932651962073867282079898040, 16.72719791907581401888377677121, 17.39898223548043825417124756271, 18.18515541879072685267669119350, 19.09932609718159268936119081654, 19.7101800633422944327428355046, 20.89170458369392278809150866131

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