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Properties

Label 1-42e2-1764.367-r0-0-0
Degree $1$
Conductor $1764$
Sign $-0.440 + 0.897i$
Analytic cond. $8.19198$
Root an. cond. $8.19198$
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  + (0.733 − 0.680i)5-s + (−0.826 + 0.563i)11-s + (−0.826 + 0.563i)13-s + (0.988 + 0.149i)17-s + (−0.5 + 0.866i)19-s + (−0.365 + 0.930i)23-s + (0.0747 − 0.997i)25-s + (−0.988 − 0.149i)29-s + 31-s + (0.365 + 0.930i)37-s + (−0.955 − 0.294i)41-s + (−0.955 + 0.294i)43-s + (−0.900 − 0.433i)47-s + (0.365 − 0.930i)53-s + (−0.222 + 0.974i)55-s + ⋯
L(s)  = 1  + (0.733 − 0.680i)5-s + (−0.826 + 0.563i)11-s + (−0.826 + 0.563i)13-s + (0.988 + 0.149i)17-s + (−0.5 + 0.866i)19-s + (−0.365 + 0.930i)23-s + (0.0747 − 0.997i)25-s + (−0.988 − 0.149i)29-s + 31-s + (0.365 + 0.930i)37-s + (−0.955 − 0.294i)41-s + (−0.955 + 0.294i)43-s + (−0.900 − 0.433i)47-s + (0.365 − 0.930i)53-s + (−0.222 + 0.974i)55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.440 + 0.897i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.440 + 0.897i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.440 + 0.897i$
Analytic conductor: \(8.19198\)
Root analytic conductor: \(8.19198\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1764} (367, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1764,\ (0:\ ),\ -0.440 + 0.897i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4321386155 + 0.6932135604i\)
\(L(\frac12)\) \(\approx\) \(0.4321386155 + 0.6932135604i\)
\(L(1)\) \(\approx\) \(0.9396491493 + 0.08758872536i\)
\(L(1)\) \(\approx\) \(0.9396491493 + 0.08758872536i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + (0.733 - 0.680i)T \)
11 \( 1 + (-0.826 + 0.563i)T \)
13 \( 1 + (-0.826 + 0.563i)T \)
17 \( 1 + (0.988 + 0.149i)T \)
19 \( 1 + (-0.5 + 0.866i)T \)
23 \( 1 + (-0.365 + 0.930i)T \)
29 \( 1 + (-0.988 - 0.149i)T \)
31 \( 1 + T \)
37 \( 1 + (0.365 + 0.930i)T \)
41 \( 1 + (-0.955 - 0.294i)T \)
43 \( 1 + (-0.955 + 0.294i)T \)
47 \( 1 + (-0.900 - 0.433i)T \)
53 \( 1 + (0.365 - 0.930i)T \)
59 \( 1 + (-0.222 + 0.974i)T \)
61 \( 1 + (-0.623 + 0.781i)T \)
67 \( 1 - T \)
71 \( 1 + (-0.623 - 0.781i)T \)
73 \( 1 + (-0.826 - 0.563i)T \)
79 \( 1 - T \)
83 \( 1 + (0.826 + 0.563i)T \)
89 \( 1 + (-0.0747 + 0.997i)T \)
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

−20.064496417584563659707408663466, −19.04205968355854524974415017904, −18.59318559165628592860433418469, −17.829868122004272215618750098084, −17.11620063955174261422664035557, −16.42421706685703478369988580605, −15.433313666746099426571403137759, −14.77415320043851819941330339613, −14.11745307368571698904209949055, −13.28194572135024523018092042837, −12.70798618373631391882454565615, −11.69498714149091201858750109536, −10.81144920438421278223147111416, −10.22667753915235469093613786651, −9.608134089036039368750622564219, −8.5609861784424672806718728484, −7.73299029272458369968451380818, −6.97397097906911567808809696344, −6.05310105113904172507240166326, −5.3988258162379577452852881617, −4.55122145975462428096226476100, −3.1551347353132067086889787111, −2.74123556377266760935768205364, −1.734873786411848018840450163900, −0.26418416868983288887189879250, 1.42732413206614451738136095406, 2.04270439386834671492642552316, 3.110305014609125740113840971029, 4.26270393276174413641147426541, 5.06224893386552996101451053902, 5.68649553742312635788070557037, 6.60049330535925848580806323252, 7.64972456368550741611254283293, 8.22677533822420184499117314344, 9.26992389213244521740130416385, 10.00888493193267675438761083280, 10.319483054082877112902749245622, 11.8284654944490230293118643614, 12.13858781781811978015299511996, 13.19711342341743692699378712696, 13.56019139938842723233389219295, 14.66754925323345345780445635094, 15.13187000187775652676837067592, 16.33414047852249519740309476279, 16.73753219234476575256279485764, 17.47363722017648798490335769630, 18.228171007565713963843775801213, 18.9878888323627823060328804623, 19.781788931215646853759246512965, 20.67333313786632528587290641665

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