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Properties

Label 1-1860-1860.239-r1-0-0
Degree $1$
Conductor $1860$
Sign $-0.704 - 0.709i$
Analytic cond. $199.884$
Root an. cond. $199.884$
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  + (0.669 − 0.743i)7-s + (0.978 + 0.207i)11-s + (−0.104 − 0.994i)13-s + (0.978 − 0.207i)17-s + (0.104 − 0.994i)19-s + (0.309 − 0.951i)23-s + (−0.809 − 0.587i)29-s + (−0.5 − 0.866i)37-s + (−0.913 + 0.406i)41-s + (0.104 − 0.994i)43-s + (0.809 − 0.587i)47-s + (−0.104 − 0.994i)49-s + (−0.669 − 0.743i)53-s + (0.913 + 0.406i)59-s − 61-s + ⋯
L(s)  = 1  + (0.669 − 0.743i)7-s + (0.978 + 0.207i)11-s + (−0.104 − 0.994i)13-s + (0.978 − 0.207i)17-s + (0.104 − 0.994i)19-s + (0.309 − 0.951i)23-s + (−0.809 − 0.587i)29-s + (−0.5 − 0.866i)37-s + (−0.913 + 0.406i)41-s + (0.104 − 0.994i)43-s + (0.809 − 0.587i)47-s + (−0.104 − 0.994i)49-s + (−0.669 − 0.743i)53-s + (0.913 + 0.406i)59-s − 61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1860 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.704 - 0.709i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1860 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.704 - 0.709i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1860\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 31\)
Sign: $-0.704 - 0.709i$
Analytic conductor: \(199.884\)
Root analytic conductor: \(199.884\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1860} (239, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1860,\ (1:\ ),\ -0.704 - 0.709i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8475566228 - 2.037160239i\)
\(L(\frac12)\) \(\approx\) \(0.8475566228 - 2.037160239i\)
\(L(1)\) \(\approx\) \(1.150850110 - 0.3879608970i\)
\(L(1)\) \(\approx\) \(1.150850110 - 0.3879608970i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 \)
31 \( 1 \)
good7 \( 1 + (0.669 - 0.743i)T \)
11 \( 1 + (0.978 + 0.207i)T \)
13 \( 1 + (-0.104 - 0.994i)T \)
17 \( 1 + (0.978 - 0.207i)T \)
19 \( 1 + (0.104 - 0.994i)T \)
23 \( 1 + (0.309 - 0.951i)T \)
29 \( 1 + (-0.809 - 0.587i)T \)
37 \( 1 + (-0.5 - 0.866i)T \)
41 \( 1 + (-0.913 + 0.406i)T \)
43 \( 1 + (0.104 - 0.994i)T \)
47 \( 1 + (0.809 - 0.587i)T \)
53 \( 1 + (-0.669 - 0.743i)T \)
59 \( 1 + (0.913 + 0.406i)T \)
61 \( 1 - T \)
67 \( 1 + (-0.5 + 0.866i)T \)
71 \( 1 + (0.669 + 0.743i)T \)
73 \( 1 + (-0.978 - 0.207i)T \)
79 \( 1 + (-0.978 + 0.207i)T \)
83 \( 1 + (0.913 - 0.406i)T \)
89 \( 1 + (0.309 + 0.951i)T \)
97 \( 1 + (-0.309 - 0.951i)T \)
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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

−20.3185872077937559848000284682, −19.19936946035802049724538135387, −18.89580402450302507976316882148, −18.109360023843064704056781124355, −17.093371372417622729457788687403, −16.74784912806724037097129548623, −15.7873781717000317616608622933, −14.905707954641783463377355950089, −14.362202868397002736404614883455, −13.77958270623947818726819529029, −12.59947064166062296807704194593, −11.90512699236060182286138757923, −11.49771153917450051524146300968, −10.52567843856882036018267825618, −9.47976489281622826365766078116, −9.02477420717754868241758903815, −8.10144042687648743870529531616, −7.36782655191610763013430698294, −6.34952398415954553872237273743, −5.65539314627746190003767276348, −4.81594480102735261781871129232, −3.84914559626082393953823175209, −3.07730104775490668383363805239, −1.699077788840542689265335721604, −1.39132261460454837958152576777, 0.40187189020211962489733363242, 1.14006906606316015558330215219, 2.21779844738928551093111995255, 3.31967940722832478825809389941, 4.10498677639482522952510052977, 4.974542591153334697601784462263, 5.73009918660949785956022508861, 6.88230708533588252629808224093, 7.39407113084325404867653003783, 8.28192979785112975365815040863, 9.06174409849277453112864695418, 10.015987923710620485343636546493, 10.63160980908513392726873661958, 11.468871213791736633797126989322, 12.1472101017634028548148639592, 13.04582280200442289378716070749, 13.783208205372330261265826750802, 14.586661993475053458717561178308, 15.03918702969592627588661543844, 16.04112990658408431296949989100, 16.982284137889972443976121173013, 17.30216066382075130207548066044, 18.1161569822359328414532944502, 18.96001817695623475708519168909, 19.78205580857800061685970419923

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