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Properties

Label 2-1224-8.5-c1-0-27
Degree $2$
Conductor $1224$
Sign $0.997 + 0.0641i$
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  + (0.733 − 1.20i)2-s + (−0.924 − 1.77i)4-s + 1.12i·5-s + 1.74·7-s + (−2.82 − 0.181i)8-s + (1.36 + 0.826i)10-s + 5.05i·11-s + 3.09i·13-s + (1.28 − 2.11i)14-s + (−2.28 + 3.28i)16-s − 17-s + 1.04i·19-s + (2.00 − 1.04i)20-s + (6.11 + 3.70i)22-s + 7.54·23-s + ⋯
L(s)  = 1  + (0.518 − 0.855i)2-s + (−0.462 − 0.886i)4-s + 0.504i·5-s + 0.660·7-s + (−0.997 − 0.0641i)8-s + (0.431 + 0.261i)10-s + 1.52i·11-s + 0.859i·13-s + (0.342 − 0.564i)14-s + (−0.572 + 0.820i)16-s − 0.242·17-s + 0.239i·19-s + (0.447 − 0.233i)20-s + (1.30 + 0.789i)22-s + 1.57·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0641i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1224 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.997 + 0.0641i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1224\)    =    \(2^{3} \cdot 3^{2} \cdot 17\)
Sign: $0.997 + 0.0641i$
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: \((2,\ 1224,\ (\ :1/2),\ 0.997 + 0.0641i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.030128494\)
\(L(\frac12)\) \(\approx\) \(2.030128494\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.733 + 1.20i)T \)
3 \( 1 \)
17 \( 1 + T \)
good5 \( 1 - 1.12iT - 5T^{2} \)
7 \( 1 - 1.74T + 7T^{2} \)
11 \( 1 - 5.05iT - 11T^{2} \)
13 \( 1 - 3.09iT - 13T^{2} \)
19 \( 1 - 1.04iT - 19T^{2} \)
23 \( 1 - 7.54T + 23T^{2} \)
29 \( 1 + 1.12iT - 29T^{2} \)
31 \( 1 + 2.68T + 31T^{2} \)
37 \( 1 - 9.14iT - 37T^{2} \)
41 \( 1 - 4.72T + 41T^{2} \)
43 \( 1 - 1.52iT - 43T^{2} \)
47 \( 1 + 7.66T + 47T^{2} \)
53 \( 1 + 3.90iT - 53T^{2} \)
59 \( 1 + 11.0iT - 59T^{2} \)
61 \( 1 - 1.12iT - 61T^{2} \)
67 \( 1 - 2.86iT - 67T^{2} \)
71 \( 1 - 3.01T + 71T^{2} \)
73 \( 1 - 12.2T + 73T^{2} \)
79 \( 1 - 3.95T + 79T^{2} \)
83 \( 1 + 9.34iT - 83T^{2} \)
89 \( 1 - 18.2T + 89T^{2} \)
97 \( 1 + 1.13T + 97T^{2} \)
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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

−9.766842282432720889379488558815, −9.252072917904000689397552552928, −8.176879354913449446811937297848, −7.01526230116391041401438592240, −6.45943537274895650609848116117, −4.96454783106690129014569671858, −4.69834484784417018158669836422, −3.49867460546986418603065985608, −2.37695584665376895573436910603, −1.47497139649514274136533100732, 0.794476663834174395489451436831, 2.80430376654734776449436990998, 3.71524099829734897368707090806, 4.91196306326704500288384325054, 5.39349881241524518057102043873, 6.28626434972228263034802376838, 7.29105850784549788047941039388, 8.088386503889884056660169067087, 8.745324715734265120449127278770, 9.276773026915773920080130780400

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