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Properties

Label 2-1224-1224.475-c0-0-0
Degree $2$
Conductor $1224$
Sign $0.766 + 0.642i$
Analytic cond. $0.610855$
Root an. cond. $0.781572$
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)2-s + (−0.5 + 0.866i)3-s + (−0.499 − 0.866i)4-s + (0.499 + 0.866i)6-s − 0.999·8-s + (−0.499 − 0.866i)9-s + (1.5 + 0.866i)11-s + 0.999·12-s + (−0.5 + 0.866i)16-s + (0.5 − 0.866i)17-s − 0.999·18-s + 19-s + (1.5 − 0.866i)22-s + (0.5 − 0.866i)24-s + (0.5 − 0.866i)25-s + ⋯
L(s)  = 1  + (0.5 − 0.866i)2-s + (−0.5 + 0.866i)3-s + (−0.499 − 0.866i)4-s + (0.499 + 0.866i)6-s − 0.999·8-s + (−0.499 − 0.866i)9-s + (1.5 + 0.866i)11-s + 0.999·12-s + (−0.5 + 0.866i)16-s + (0.5 − 0.866i)17-s − 0.999·18-s + 19-s + (1.5 − 0.866i)22-s + (0.5 − 0.866i)24-s + (0.5 − 0.866i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 + 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 + 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1224\)    =    \(2^{3} \cdot 3^{2} \cdot 17\)
Sign: $0.766 + 0.642i$
Analytic conductor: \(0.610855\)
Root analytic conductor: \(0.781572\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1224} (475, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1224,\ (\ :0),\ 0.766 + 0.642i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.164082055\)
\(L(\frac12)\) \(\approx\) \(1.164082055\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 + 0.866i)T \)
3 \( 1 + (0.5 - 0.866i)T \)
17 \( 1 + (-0.5 + 0.866i)T \)
good5 \( 1 + (-0.5 + 0.866i)T^{2} \)
7 \( 1 + (-0.5 - 0.866i)T^{2} \)
11 \( 1 + (-1.5 - 0.866i)T + (0.5 + 0.866i)T^{2} \)
13 \( 1 + (0.5 - 0.866i)T^{2} \)
19 \( 1 - T + T^{2} \)
23 \( 1 + (-0.5 + 0.866i)T^{2} \)
29 \( 1 + (-0.5 - 0.866i)T^{2} \)
31 \( 1 + (-0.5 + 0.866i)T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 + (1.5 - 0.866i)T + (0.5 - 0.866i)T^{2} \)
43 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
47 \( 1 + (0.5 + 0.866i)T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
61 \( 1 + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + 1.73iT - T^{2} \)
79 \( 1 + (-0.5 - 0.866i)T^{2} \)
83 \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \)
89 \( 1 + 2T + T^{2} \)
97 \( 1 + (1.5 + 0.866i)T + (0.5 + 0.866i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.824206081209817201335451375499, −9.454019278902760529682705033498, −8.650530414201997835831845181735, −7.10410753676822757681366670855, −6.26620383264113419268109010663, −5.30198964224800078780323840239, −4.56589374490632506740613262124, −3.80566119355987164770972951055, −2.83981438972149856427465875847, −1.23478224199790350519964995690, 1.32951174616103366880323886147, 3.10861402964011197217629496729, 4.00971955490330112688861728773, 5.31806281742305860886591632788, 5.85438447446524497204034281352, 6.72965927994260614838719887549, 7.24775440333424610812902713427, 8.290816464926837952884566588120, 8.801762741230841419501987607891, 9.866397903838439585928541953610

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