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Properties

Label 1-1224-1224.115-r1-0-0
Degree $1$
Conductor $1224$
Sign $0.978 + 0.208i$
Analytic cond. $131.537$
Root an. cond. $131.537$
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 + (0.866 − 0.5i)7-s + (−0.866 + 0.5i)11-s + (0.5 − 0.866i)13-s − 19-s + (−0.866 − 0.5i)23-s + (0.5 + 0.866i)25-s + (−0.866 + 0.5i)29-s + (0.866 + 0.5i)31-s + 35-s − i·37-s + (0.866 + 0.5i)41-s + (0.5 + 0.866i)43-s + (0.5 + 0.866i)47-s + (0.5 − 0.866i)49-s + ⋯
L(s)  = 1  + (0.866 + 0.5i)5-s + (0.866 − 0.5i)7-s + (−0.866 + 0.5i)11-s + (0.5 − 0.866i)13-s − 19-s + (−0.866 − 0.5i)23-s + (0.5 + 0.866i)25-s + (−0.866 + 0.5i)29-s + (0.866 + 0.5i)31-s + 35-s − i·37-s + (0.866 + 0.5i)41-s + (0.5 + 0.866i)43-s + (0.5 + 0.866i)47-s + (0.5 − 0.866i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1224 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.978 + 0.208i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1224 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.978 + 0.208i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1224\)    =    \(2^{3} \cdot 3^{2} \cdot 17\)
Sign: $0.978 + 0.208i$
Analytic conductor: \(131.537\)
Root analytic conductor: \(131.537\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1224} (115, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1224,\ (1:\ ),\ 0.978 + 0.208i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.798300544 + 0.2945606635i\)
\(L(\frac12)\) \(\approx\) \(2.798300544 + 0.2945606635i\)
\(L(1)\) \(\approx\) \(1.358533282 + 0.05639667377i\)
\(L(1)\) \(\approx\) \(1.358533282 + 0.05639667377i\)

Euler product

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

−21.0039862028337961157684386687, −20.48923758425219752607412353953, −19.20440380908871605834276818391, −18.575555472251404891769211952566, −17.85392389974810235951941403043, −17.127065455098607650250356608475, −16.398496060913522536013269179814, −15.51874843562587973003401444645, −14.73387323330865220311342947901, −13.68684709697803359408035468004, −13.44327128061233346687413907327, −12.30598057594098348486133804595, −11.56933260100572371223226789764, −10.6990256210132382342243283335, −9.90563354250922096192841814594, −8.86045791092485579772106900969, −8.44932643045517234049503076272, −7.466460977741359774774344676075, −6.16066708309628454864447668395, −5.69322232126926307724051484589, −4.77005892111015853546319355877, −3.91219074268907821904183264755, −2.37502777395386831950015221733, −1.944350113060143009069708384272, −0.71882434016470952134530089446, 0.780839732036449389254985368553, 1.94383671266559338644733268323, 2.620012951132274440328905861156, 3.849845680831934806172520319050, 4.83461231739376616417695079394, 5.65944930351439212205529119437, 6.46853504427137356002660966529, 7.51800795526031553562439867962, 8.12356435594172741332399959354, 9.14139595955630870308295736622, 10.295370276235626252598803176131, 10.5487998123219285095219727050, 11.358377709797956933297987943987, 12.68682222545518714165853981847, 13.126880380816920742176868232206, 14.14603248239127654872298014530, 14.63342036110666416175924636266, 15.48167704120830516422617418831, 16.40478409465933409315939381830, 17.44370054560201436019459501718, 17.84699716889895639592231360152, 18.40508250379116125713445727159, 19.444087202401256361349401660884, 20.47889415489060497273911581521, 20.91260758209767512721071823248

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