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Properties

Label 2-1224-136.101-c1-0-46
Degree $2$
Conductor $1224$
Sign $0.00613 - 0.999i$
Analytic cond. $9.77368$
Root an. cond. $3.12629$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.0535 + 1.41i)2-s + (−1.99 − 0.151i)4-s + 3.17·5-s + 2.51i·7-s + (0.320 − 2.81i)8-s + (−0.170 + 4.48i)10-s + 5.40·11-s + 3.61i·13-s + (−3.55 − 0.134i)14-s + (3.95 + 0.604i)16-s + (4.09 − 0.442i)17-s − 6.87i·19-s + (−6.32 − 0.480i)20-s + (−0.289 + 7.63i)22-s − 7.54i·23-s + ⋯
L(s)  = 1  + (−0.0378 + 0.999i)2-s + (−0.997 − 0.0757i)4-s + 1.41·5-s + 0.950i·7-s + (0.113 − 0.993i)8-s + (−0.0537 + 1.41i)10-s + 1.62·11-s + 1.00i·13-s + (−0.950 − 0.0360i)14-s + (0.988 + 0.151i)16-s + (0.994 − 0.107i)17-s − 1.57i·19-s + (−1.41 − 0.107i)20-s + (−0.0616 + 1.62i)22-s − 1.57i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.00613 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1224 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.00613 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1224\)    =    \(2^{3} \cdot 3^{2} \cdot 17\)
Sign: $0.00613 - 0.999i$
Analytic conductor: \(9.77368\)
Root analytic conductor: \(3.12629\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1224} (1189, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1224,\ (\ :1/2),\ 0.00613 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.104136351\)
\(L(\frac12)\) \(\approx\) \(2.104136351\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.0535 - 1.41i)T \)
3 \( 1 \)
17 \( 1 + (-4.09 + 0.442i)T \)
good5 \( 1 - 3.17T + 5T^{2} \)
7 \( 1 - 2.51iT - 7T^{2} \)
11 \( 1 - 5.40T + 11T^{2} \)
13 \( 1 - 3.61iT - 13T^{2} \)
19 \( 1 + 6.87iT - 19T^{2} \)
23 \( 1 + 7.54iT - 23T^{2} \)
29 \( 1 + 3.32T + 29T^{2} \)
31 \( 1 + 1.30iT - 31T^{2} \)
37 \( 1 + 6.65T + 37T^{2} \)
41 \( 1 - 5.04iT - 41T^{2} \)
43 \( 1 - 5.85iT - 43T^{2} \)
47 \( 1 - 8.02T + 47T^{2} \)
53 \( 1 - 7.67iT - 53T^{2} \)
59 \( 1 + 4.23iT - 59T^{2} \)
61 \( 1 + 3.88T + 61T^{2} \)
67 \( 1 - 10.8iT - 67T^{2} \)
71 \( 1 - 4.23iT - 71T^{2} \)
73 \( 1 + 0.674iT - 73T^{2} \)
79 \( 1 - 1.43iT - 79T^{2} \)
83 \( 1 + 16.1iT - 83T^{2} \)
89 \( 1 + 9.77T + 89T^{2} \)
97 \( 1 + 0.954iT - 97T^{2} \)
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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

−9.397556236642408969500842509673, −9.194933736409888118195694273740, −8.574913454565965686351914669409, −7.16402845033408987271518146305, −6.46895927384375199957976337695, −5.96607649428164212234986528996, −5.06894587217374797174599727307, −4.15635543826667498654138355711, −2.63959332976634910062877693883, −1.34846178451999697279671966892, 1.18306936576863847619699125484, 1.81623638217951229401406610885, 3.43567582120266322221596719305, 3.88708746048428167482953540274, 5.42791301600297528611636443845, 5.80132768144355970857603234474, 7.09656834028383250872471737251, 8.048516287924749615065376550781, 9.112807752492029442868731755056, 9.713105748506745674322356736686

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