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Properties

Label 1-1840-1840.1533-r0-0-0
Degree $1$
Conductor $1840$
Sign $-0.444 - 0.895i$
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  + (0.654 − 0.755i)3-s + (0.909 − 0.415i)7-s + (−0.142 − 0.989i)9-s + (−0.281 − 0.959i)11-s + (−0.415 + 0.909i)13-s + (0.540 + 0.841i)17-s + (−0.540 + 0.841i)19-s + (0.281 − 0.959i)21-s + (−0.841 − 0.540i)27-s + (−0.540 − 0.841i)29-s + (−0.654 − 0.755i)31-s + (−0.909 − 0.415i)33-s + (−0.142 − 0.989i)37-s + (0.415 + 0.909i)39-s + (0.142 − 0.989i)41-s + ⋯
L(s)  = 1  + (0.654 − 0.755i)3-s + (0.909 − 0.415i)7-s + (−0.142 − 0.989i)9-s + (−0.281 − 0.959i)11-s + (−0.415 + 0.909i)13-s + (0.540 + 0.841i)17-s + (−0.540 + 0.841i)19-s + (0.281 − 0.959i)21-s + (−0.841 − 0.540i)27-s + (−0.540 − 0.841i)29-s + (−0.654 − 0.755i)31-s + (−0.909 − 0.415i)33-s + (−0.142 − 0.989i)37-s + (0.415 + 0.909i)39-s + (0.142 − 0.989i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.444 - 0.895i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.444 - 0.895i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1840\)    =    \(2^{4} \cdot 5 \cdot 23\)
Sign: $-0.444 - 0.895i$
Analytic conductor: \(8.54492\)
Root analytic conductor: \(8.54492\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1840} (1533, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1840,\ (0:\ ),\ -0.444 - 0.895i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.004992237 - 1.619825349i\)
\(L(\frac12)\) \(\approx\) \(1.004992237 - 1.619825349i\)
\(L(1)\) \(\approx\) \(1.205947708 - 0.5676643759i\)
\(L(1)\) \(\approx\) \(1.205947708 - 0.5676643759i\)

Euler product

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

−20.27207921948057244895780877717, −20.02609115623605047019366804301, −18.937803594234047806022686851674, −18.123412256474423305201370446925, −17.514055482582227038925291691910, −16.66407358209672118792674845149, −15.713907971901890688833170111737, −15.197769234393614456735589233551, −14.60171916082263197393223807649, −13.98516668732091710202782080770, −12.91348135410619619140693891707, −12.3141642755483822525556040328, −11.154706537179074848543949983049, −10.71459163236562031401325979908, −9.66670143086291774983519621312, −9.25508468364567455528286078095, −8.15680227234150262566554667116, −7.76437159541572251702790685495, −6.8007773879447576682502688490, −5.25775470044485910909582119317, −5.07345902657477603065419902142, −4.18743273524790262477762463147, −2.990475885918970397913608810392, −2.46602200803689631129999066221, −1.38480766878040931578251685938, 0.59186781931837991491793574122, 1.806292635847709890953223900945, 2.22545908777551611237136168758, 3.660681555319426233376316926300, 4.03814953793184672520137535692, 5.43400043104083431650456859516, 6.1112665696907615478747732487, 7.15146515459330920366907724331, 7.78485872464151846860695921761, 8.42563994706095114442082678563, 9.11716787433984653501360939974, 10.16213044251366229671404588250, 11.014430863399106848741321998506, 11.77294524915237468938580789301, 12.51590762779425408629323651636, 13.357542330767423021736317450339, 14.05068123307285421339462369875, 14.55728352263507616879470045527, 15.20259021421366347161895942518, 16.42636935861879076584681994103, 17.030521042329130404961660712981, 17.739878538808011629044122568099, 18.75318946972791679192470752577, 18.997131258237608022704703291913, 19.779102868418894590419253164075

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