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Properties

Label 1-1840-1840.419-r0-0-0
Degree $1$
Conductor $1840$
Sign $0.999 + 0.0182i$
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  + (0.989 − 0.142i)3-s + (−0.654 + 0.755i)7-s + (0.959 − 0.281i)9-s + (−0.540 − 0.841i)11-s + (−0.755 + 0.654i)13-s + (0.415 − 0.909i)17-s + (−0.909 + 0.415i)19-s + (−0.540 + 0.841i)21-s + (0.909 − 0.415i)27-s + (0.909 + 0.415i)29-s + (0.142 − 0.989i)31-s + (−0.654 − 0.755i)33-s + (0.281 + 0.959i)37-s + (−0.654 + 0.755i)39-s + (0.959 + 0.281i)41-s + ⋯
L(s)  = 1  + (0.989 − 0.142i)3-s + (−0.654 + 0.755i)7-s + (0.959 − 0.281i)9-s + (−0.540 − 0.841i)11-s + (−0.755 + 0.654i)13-s + (0.415 − 0.909i)17-s + (−0.909 + 0.415i)19-s + (−0.540 + 0.841i)21-s + (0.909 − 0.415i)27-s + (0.909 + 0.415i)29-s + (0.142 − 0.989i)31-s + (−0.654 − 0.755i)33-s + (0.281 + 0.959i)37-s + (−0.654 + 0.755i)39-s + (0.959 + 0.281i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0182i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0182i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1840\)    =    \(2^{4} \cdot 5 \cdot 23\)
Sign: $0.999 + 0.0182i$
Analytic conductor: \(8.54492\)
Root analytic conductor: \(8.54492\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1840} (419, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1840,\ (0:\ ),\ 0.999 + 0.0182i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.059596962 + 0.01883188772i\)
\(L(\frac12)\) \(\approx\) \(2.059596962 + 0.01883188772i\)
\(L(1)\) \(\approx\) \(1.370169786 + 0.01219062825i\)
\(L(1)\) \(\approx\) \(1.370169786 + 0.01219062825i\)

Euler product

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

−19.94833745844250035960318957277, −19.55527808580021979732825704010, −19.03069455839829849708868392806, −17.803344726854737095261524139919, −17.36038273891829846464400024212, −16.30634297140134013525117130004, −15.64705892039708622572143396450, −14.92776389232482591602406598098, −14.33049807265993090399658392735, −13.46960551931994314835284988959, −12.65281328763793755678487183221, −12.46651326707851242115056852236, −10.77878763749955432264602942766, −10.28554989823687837145420658315, −9.7281219883481149588874253733, −8.80673338808471457717526375451, −7.95015833918696994589268946601, −7.329714559644871220694607036963, −6.62525961957549728839692821012, −5.42421782892997813571823625111, −4.38340428393994217648730408314, −3.84923126740103823075972258653, −2.75941667634542172909330889461, −2.20770937994339966257618234560, −0.855802854846433120065996712376, 0.85323977922395051273632063462, 2.37502980775189571133525560471, 2.60326938082682405188013810189, 3.62275843343615152657901772985, 4.53797629912194665716094626955, 5.58345323762032347305223425144, 6.45547188759550186843696993737, 7.263972628703529484563394366338, 8.14717640054890333404275285062, 8.777462746889468691173656451466, 9.55491526467133891827883889503, 10.09595207671142816912720581856, 11.21977274134376987392215022187, 12.18422191765601518955711888038, 12.718198192775774312100298381241, 13.57271570215333903195766229311, 14.206767446838359708345631830267, 14.919671193503125425176536232046, 15.73639313836172222003695740715, 16.25580988985009936251874141856, 17.13600897749939647674550403513, 18.33215832704355516004859714239, 18.83516724120074250886759427025, 19.23842354947788492308498586746, 20.062031467819529968120859195335

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