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Properties

Label 1-1872-1872.565-r1-0-0
Degree $1$
Conductor $1872$
Sign $0.427 - 0.904i$
Analytic cond. $201.174$
Root an. cond. $201.174$
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  + (−0.5 + 0.866i)5-s − i·7-s + (0.5 − 0.866i)11-s + (0.5 − 0.866i)17-s + (0.5 − 0.866i)19-s + 23-s + (−0.5 − 0.866i)25-s + (0.866 + 0.5i)29-s + (0.866 + 0.5i)31-s + (0.866 + 0.5i)35-s + (0.5 + 0.866i)37-s − i·41-s − i·43-s + (0.866 − 0.5i)47-s − 49-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)5-s − i·7-s + (0.5 − 0.866i)11-s + (0.5 − 0.866i)17-s + (0.5 − 0.866i)19-s + 23-s + (−0.5 − 0.866i)25-s + (0.866 + 0.5i)29-s + (0.866 + 0.5i)31-s + (0.866 + 0.5i)35-s + (0.5 + 0.866i)37-s − i·41-s − i·43-s + (0.866 − 0.5i)47-s − 49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.427 - 0.904i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.427 - 0.904i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1872\)    =    \(2^{4} \cdot 3^{2} \cdot 13\)
Sign: $0.427 - 0.904i$
Analytic conductor: \(201.174\)
Root analytic conductor: \(201.174\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1872} (565, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1872,\ (1:\ ),\ 0.427 - 0.904i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.872582478 - 1.186370035i\)
\(L(\frac12)\) \(\approx\) \(1.872582478 - 1.186370035i\)
\(L(1)\) \(\approx\) \(1.109063408 - 0.1499887737i\)
\(L(1)\) \(\approx\) \(1.109063408 - 0.1499887737i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
13 \( 1 \)
good5 \( 1 + (-0.5 + 0.866i)T \)
7 \( 1 - iT \)
11 \( 1 + (0.5 - 0.866i)T \)
17 \( 1 + (0.5 - 0.866i)T \)
19 \( 1 + (0.5 - 0.866i)T \)
23 \( 1 + T \)
29 \( 1 + (0.866 + 0.5i)T \)
31 \( 1 + (0.866 + 0.5i)T \)
37 \( 1 + (0.5 + 0.866i)T \)
41 \( 1 - iT \)
43 \( 1 - iT \)
47 \( 1 + (0.866 - 0.5i)T \)
53 \( 1 + iT \)
59 \( 1 + (0.5 + 0.866i)T \)
61 \( 1 - iT \)
67 \( 1 - T \)
71 \( 1 + (-0.866 - 0.5i)T \)
73 \( 1 - iT \)
79 \( 1 + (-0.5 - 0.866i)T \)
83 \( 1 + (0.5 + 0.866i)T \)
89 \( 1 + (-0.866 + 0.5i)T \)
97 \( 1 - iT \)
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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

−19.99567249280141874414216404404, −19.27978722336585044298103391345, −18.80497420481420529672822474335, −17.69385615308713482843867035120, −17.19924287074087466893865522725, −16.30344318945794763141350371555, −15.671002357126940290169338184921, −14.95090374054638147572503310608, −14.35279760672965399187821147344, −13.14106183654764783937838027577, −12.51181111518029260366577457592, −12.0140339441041185538415411410, −11.35562565091755755695053044074, −10.14336245504300455416874281724, −9.46656290165460152421108723203, −8.70152267461749972309960825197, −8.06211240108501846491004923044, −7.24395822840686180056870531645, −6.12333421202976609501946023511, −5.46310472292378029124257731106, −4.562062282780395101144238294115, −3.86582375096224563689816696620, −2.76520729257886950450734842219, −1.72304622530473126459061636982, −0.89369425806819236570670052194, 0.52785061196544538446229395807, 1.18298476620373362186864732512, 2.90643916405267907737042448979, 3.15203159222415639343728482174, 4.23985787625133555134321782013, 5.0054040735009419854485064753, 6.22348707986122816344506673542, 6.93845558722591839902847031571, 7.44975972534828402448165649484, 8.37797571428731143248958791114, 9.26988316308070201686841607884, 10.23090036198964538081961438072, 10.82083629951700165561495256536, 11.53751403816523043522740236548, 12.12281587343612736317779251663, 13.5077540598746706969359994019, 13.77208124122614827699442733499, 14.572048547576653670867587122208, 15.366472001336299054884511286623, 16.16892574282839862456556048025, 16.800162048490000691067846604309, 17.62222155878457320499511867848, 18.39944846463117183283105313486, 19.12463816014482916268155906030, 19.71654255100553870381151094484

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