↑

Properties

Label 1-1840-1840.339-r0-0-0
Degree $1$
Conductor $1840$
Sign $-0.998 + 0.0489i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.540 + 0.841i)3-s + (−0.959 − 0.281i)7-s + (−0.415 − 0.909i)9-s + (0.755 + 0.654i)11-s + (0.281 + 0.959i)13-s + (−0.142 − 0.989i)17-s + (−0.989 − 0.142i)19-s + (0.755 − 0.654i)21-s + (0.989 + 0.142i)27-s + (0.989 − 0.142i)29-s + (−0.841 + 0.540i)31-s + (−0.959 + 0.281i)33-s + (0.909 − 0.415i)37-s + (−0.959 − 0.281i)39-s + (−0.415 + 0.909i)41-s + ⋯
L(s)  = 1  + (−0.540 + 0.841i)3-s + (−0.959 − 0.281i)7-s + (−0.415 − 0.909i)9-s + (0.755 + 0.654i)11-s + (0.281 + 0.959i)13-s + (−0.142 − 0.989i)17-s + (−0.989 − 0.142i)19-s + (0.755 − 0.654i)21-s + (0.989 + 0.142i)27-s + (0.989 − 0.142i)29-s + (−0.841 + 0.540i)31-s + (−0.959 + 0.281i)33-s + (0.909 − 0.415i)37-s + (−0.959 − 0.281i)39-s + (−0.415 + 0.909i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.998 + 0.0489i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.998 + 0.0489i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1840\)    =    \(2^{4} \cdot 5 \cdot 23\)
Sign: $-0.998 + 0.0489i$
Analytic conductor: \(8.54492\)
Root analytic conductor: \(8.54492\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1840} (339, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1840,\ (0:\ ),\ -0.998 + 0.0489i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.01113889699 + 0.4547170216i\)
\(L(\frac12)\) \(\approx\) \(0.01113889699 + 0.4547170216i\)
\(L(1)\) \(\approx\) \(0.6655183318 + 0.2409506984i\)
\(L(1)\) \(\approx\) \(0.6655183318 + 0.2409506984i\)

Euler product

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

Imaginary part of the first few zeros on the critical line

−19.58243414109431000791720324148, −19.06166610748593082384447364186, −18.41911071460120917216043719434, −17.535057208381136572022827203201, −16.90239272699103096969983493697, −16.32526837287498621101192025013, −15.37804032040445250756370564207, −14.629187507952619947154582771199, −13.52968269256340578730402321919, −13.10630613097114290585705309968, −12.358376756775302012149940842598, −11.77552400042562753114200001029, −10.69506120934346963941202088838, −10.32387363092566305112144122206, −8.98369872480732239430756562394, −8.45862300042234164245366212233, −7.53071568471737986092036704491, −6.5327715625806683509220166236, −6.1163873664389677363123919565, −5.46426786799541022069174742804, −4.15832578112189485951882100389, −3.270661704814175700060695765582, −2.32797528491471898494937506569, −1.28267153119580241986908000203, −0.192955310351383760956422624283, 1.19312517038083195659968642888, 2.54794151086295329580446238411, 3.52089915138418911240664923800, 4.32318610864775828331300752048, 4.8339638878500854120818632868, 6.18020953128891990037681473425, 6.512968741362631495099977768183, 7.366251126456863373481068247234, 8.8019023451428722304714373907, 9.31521403065316005775737046287, 9.92633797816618112506964741717, 10.75438997444097283807715528515, 11.52417194984632023035159953689, 12.20506564879455804357011152282, 12.98798412311351018316926118585, 13.99587482612514109742564100845, 14.650981740232221191535768624675, 15.5193841754686719949469500180, 16.203631248381155200316284621675, 16.70574821219695541513040086081, 17.38805880535112920779187271621, 18.228857609050658277598449554958, 19.083451701636821147271575215846, 19.98586226855968046289094222818, 20.356342981175330678925507068435

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