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Properties

Label 1-42e2-1764.1319-r1-0-0
Degree $1$
Conductor $1764$
Sign $0.805 + 0.592i$
Analytic cond. $189.568$
Root an. cond. $189.568$
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  + (−0.900 − 0.433i)5-s + (0.623 − 0.781i)11-s + (0.988 + 0.149i)13-s + (0.955 − 0.294i)17-s + (−0.5 + 0.866i)19-s + (−0.222 − 0.974i)23-s + (0.623 + 0.781i)25-s + (0.733 + 0.680i)29-s + (−0.5 + 0.866i)31-s + (−0.733 − 0.680i)37-s + (0.0747 + 0.997i)41-s + (−0.0747 + 0.997i)43-s + (0.988 + 0.149i)47-s + (0.733 − 0.680i)53-s + (−0.900 + 0.433i)55-s + ⋯
L(s)  = 1  + (−0.900 − 0.433i)5-s + (0.623 − 0.781i)11-s + (0.988 + 0.149i)13-s + (0.955 − 0.294i)17-s + (−0.5 + 0.866i)19-s + (−0.222 − 0.974i)23-s + (0.623 + 0.781i)25-s + (0.733 + 0.680i)29-s + (−0.5 + 0.866i)31-s + (−0.733 − 0.680i)37-s + (0.0747 + 0.997i)41-s + (−0.0747 + 0.997i)43-s + (0.988 + 0.149i)47-s + (0.733 − 0.680i)53-s + (−0.900 + 0.433i)55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.805 + 0.592i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.805 + 0.592i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.805 + 0.592i$
Analytic conductor: \(189.568\)
Root analytic conductor: \(189.568\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1764} (1319, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1764,\ (1:\ ),\ 0.805 + 0.592i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.673779167 + 0.5491217073i\)
\(L(\frac12)\) \(\approx\) \(1.673779167 + 0.5491217073i\)
\(L(1)\) \(\approx\) \(1.007336649 + 0.02983658738i\)
\(L(1)\) \(\approx\) \(1.007336649 + 0.02983658738i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + (-0.900 - 0.433i)T \)
11 \( 1 + (0.623 - 0.781i)T \)
13 \( 1 + (0.988 + 0.149i)T \)
17 \( 1 + (0.955 - 0.294i)T \)
19 \( 1 + (-0.5 + 0.866i)T \)
23 \( 1 + (-0.222 - 0.974i)T \)
29 \( 1 + (0.733 + 0.680i)T \)
31 \( 1 + (-0.5 + 0.866i)T \)
37 \( 1 + (-0.733 - 0.680i)T \)
41 \( 1 + (0.0747 + 0.997i)T \)
43 \( 1 + (-0.0747 + 0.997i)T \)
47 \( 1 + (0.988 + 0.149i)T \)
53 \( 1 + (0.733 - 0.680i)T \)
59 \( 1 + (-0.0747 + 0.997i)T \)
61 \( 1 + (-0.955 + 0.294i)T \)
67 \( 1 + (0.5 - 0.866i)T \)
71 \( 1 + (-0.222 - 0.974i)T \)
73 \( 1 + (-0.365 + 0.930i)T \)
79 \( 1 + (0.5 + 0.866i)T \)
83 \( 1 + (0.988 - 0.149i)T \)
89 \( 1 + (-0.988 + 0.149i)T \)
97 \( 1 + (0.5 - 0.866i)T \)
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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

−20.057929710956300788938440892982, −19.03957363086628509841055231463, −18.82921557353684090210840093554, −17.67043586543970686611627161352, −17.17473847477574333296982062032, −16.1572483275542726130126851242, −15.388428758840250713876156045695, −15.050344268265463045704560626372, −14.04321640608907052582256110386, −13.35679221217416338152454331830, −12.24682758018847991342169609456, −11.86976734872504392483608837677, −10.95275369256572799628614887909, −10.31998946152063975942115836819, −9.335175278123154874585663630130, −8.517900772602204141130901754103, −7.707265845834465874599004355379, −7.02034081105273541257907095815, −6.22164030885117503594258489954, −5.24645963263463992472683804662, −4.086067929642837213842277243836, −3.7190553545211259229798595488, −2.630995660854917819671426369616, −1.51221701128556022686123944216, −0.43335880375145574071015502396, 0.807585858135428459997306119553, 1.477653743221159855064280569201, 3.01154351855528712432558661622, 3.69796935410391310328653364961, 4.407498587355022715654069203384, 5.45440466997686330610501214820, 6.27173634337759888618106779519, 7.122105645400939718226707743382, 8.19232153841053725727333018525, 8.53236412423433478060530871302, 9.3799470887096592519815762953, 10.56865904383696976833543477565, 11.0586286304082313706867111661, 12.12458251897977628452185475794, 12.35997579862927332558555020688, 13.46774425937055177888477629522, 14.28824006715400501268846180997, 14.84480672612103027401755235242, 16.004462940179578459097814951137, 16.3375789919900207309463557339, 16.89656031858188399895931580429, 18.15593587911637320299368209918, 18.66228885308543184392659577501, 19.480916311135035666156958567845, 19.98910889751611915310924258305

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