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Properties

Label 2-1224-8.5-c1-0-31
Degree $2$
Conductor $1224$
Sign $0.787 + 0.616i$
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  + (−1.37 − 0.310i)2-s + (1.80 + 0.856i)4-s − 2.33i·5-s − 1.57·7-s + (−2.22 − 1.74i)8-s + (−0.724 + 3.22i)10-s + 2.23i·11-s + 1.35i·13-s + (2.17 + 0.490i)14-s + (2.53 + 3.09i)16-s − 17-s + 4.21i·19-s + (1.99 − 4.21i)20-s + (0.695 − 3.09i)22-s + 8.48·23-s + ⋯
L(s)  = 1  + (−0.975 − 0.219i)2-s + (0.903 + 0.428i)4-s − 1.04i·5-s − 0.597·7-s + (−0.787 − 0.616i)8-s + (−0.229 + 1.01i)10-s + 0.675i·11-s + 0.375i·13-s + (0.582 + 0.131i)14-s + (0.632 + 0.774i)16-s − 0.242·17-s + 0.967i·19-s + (0.447 − 0.943i)20-s + (0.148 − 0.658i)22-s + 1.77·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.787 + 0.616i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1224 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.787 + 0.616i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1224\)    =    \(2^{3} \cdot 3^{2} \cdot 17\)
Sign: $0.787 + 0.616i$
Analytic conductor: \(9.77368\)
Root analytic conductor: \(3.12629\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1224} (613, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1224,\ (\ :1/2),\ 0.787 + 0.616i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9628471944\)
\(L(\frac12)\) \(\approx\) \(0.9628471944\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.37 + 0.310i)T \)
3 \( 1 \)
17 \( 1 + T \)
good5 \( 1 + 2.33iT - 5T^{2} \)
7 \( 1 + 1.57T + 7T^{2} \)
11 \( 1 - 2.23iT - 11T^{2} \)
13 \( 1 - 1.35iT - 13T^{2} \)
19 \( 1 - 4.21iT - 19T^{2} \)
23 \( 1 - 8.48T + 23T^{2} \)
29 \( 1 - 2.33iT - 29T^{2} \)
31 \( 1 - 9.09T + 31T^{2} \)
37 \( 1 + 6.29iT - 37T^{2} \)
41 \( 1 - 0.550T + 41T^{2} \)
43 \( 1 + 11.6iT - 43T^{2} \)
47 \( 1 - 4.96T + 47T^{2} \)
53 \( 1 + 1.77iT - 53T^{2} \)
59 \( 1 + 3.94iT - 59T^{2} \)
61 \( 1 + 2.33iT - 61T^{2} \)
67 \( 1 + 2.44iT - 67T^{2} \)
71 \( 1 - 3.86T + 71T^{2} \)
73 \( 1 - 1.39T + 73T^{2} \)
79 \( 1 + 3.64T + 79T^{2} \)
83 \( 1 - 8.09iT - 83T^{2} \)
89 \( 1 + 6.43T + 89T^{2} \)
97 \( 1 - 11.5T + 97T^{2} \)
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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

−9.503617773306759846138113791831, −8.922442470572623181805798418982, −8.269656986992751240775525924052, −7.23945388882796188023257806298, −6.59982394891228348311236468407, −5.47004984701364441179647470339, −4.40745678510249243032113045600, −3.25747428657113696272469454462, −2.00309851616261767414780238234, −0.807790654931858911717680244497, 0.891587055197963600574124810015, 2.77776968806371675871839975976, 3.04540371334432396548250639695, 4.82208858108018776447780521676, 6.08271132332038388514108620635, 6.61490520637176777722618802624, 7.28668103754644716918690193677, 8.239148405949295132182272862219, 9.015562548815105088780263202904, 9.804373613515783734734302805816

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