Properties

Label 1-1872-1872.283-r1-0-0
Degree $1$
Conductor $1872$
Sign $-0.973 + 0.228i$
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  + (−0.866 + 0.5i)5-s − 7-s + (0.866 − 0.5i)11-s + (−0.5 − 0.866i)17-s + (−0.866 + 0.5i)19-s + 23-s + (0.5 − 0.866i)25-s + (−0.866 + 0.5i)29-s + (−0.5 − 0.866i)31-s + (0.866 − 0.5i)35-s + (0.866 + 0.5i)37-s + 41-s i·43-s + (−0.5 + 0.866i)47-s + 49-s + ⋯
L(s)  = 1  + (−0.866 + 0.5i)5-s − 7-s + (0.866 − 0.5i)11-s + (−0.5 − 0.866i)17-s + (−0.866 + 0.5i)19-s + 23-s + (0.5 − 0.866i)25-s + (−0.866 + 0.5i)29-s + (−0.5 − 0.866i)31-s + (0.866 − 0.5i)35-s + (0.866 + 0.5i)37-s + 41-s i·43-s + (−0.5 + 0.866i)47-s + 49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.973 + 0.228i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.973 + 0.228i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1872\)    =    \(2^{4} \cdot 3^{2} \cdot 13\)
Sign: $-0.973 + 0.228i$
Analytic conductor: \(201.174\)
Root analytic conductor: \(201.174\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1872} (283, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1872,\ (1:\ ),\ -0.973 + 0.228i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.009866751234 + 0.08504998648i\)
\(L(\frac12)\) \(\approx\) \(0.009866751234 + 0.08504998648i\)
\(L(1)\) \(\approx\) \(0.7489711589 + 0.01300692045i\)
\(L(1)\) \(\approx\) \(0.7489711589 + 0.01300692045i\)

Euler product

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

−19.56451292478292368183529907141, −19.22440274614820209307383344523, −18.12911505019963857749120698576, −17.187887122969370622110191117761, −16.648408297108112811939613520704, −15.98321819744982247468958170661, −15.01936046813774747290912929792, −14.8132295016979200156939833167, −13.38531626418879835020091217012, −12.81948559477689613173939749125, −12.335036537058975004063270126548, −11.3363593246550607464634741626, −10.76105339142550111794896602170, −9.599380319381540550199666076416, −9.06022591910967350080113020902, −8.33839384651585990117855305370, −7.28321716243203851324599771067, −6.69931468151019569495327991320, −5.853108037364304725482503067543, −4.67973950601570400588000160043, −4.03513999909114402821890251207, −3.32623847729911933910499779534, −2.17554854796751342938794362201, −1.02994723142888675550028006623, −0.021547032819752100598319740667, 0.84426531317757688328300291137, 2.295300820881312519879963196326, 3.21697467527236805630483980411, 3.81138622093857306344715224279, 4.65576864178374651427651753006, 5.92824661687618996191598328417, 6.58489174751042521774576154999, 7.2424403326531248016505298694, 8.10633234145268597453366964290, 9.13585563622492941884262259839, 9.54202035681682666674292788793, 10.88012570085952123390486168196, 11.1054097573114446397672474917, 12.138315121843050995710480257347, 12.75472773298757003009563645368, 13.61474386406259656138433806365, 14.46866975874427349638778750429, 15.14976586838796975977172219588, 15.8327528697469999824189681965, 16.6319816599729083977472073716, 17.07124720626599415892726965726, 18.42965083046657366951572804830, 18.78008078509739720059142899071, 19.518379667225931463153303185646, 20.051820851342571628405549217150

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