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Properties

Label 2-1224-8.5-c1-0-69
Degree $2$
Conductor $1224$
Sign $-0.366 + 0.930i$
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  + (−1.30 + 0.548i)2-s + (1.39 − 1.43i)4-s − 3.97i·5-s + 3.04·7-s + (−1.03 + 2.63i)8-s + (2.18 + 5.17i)10-s − 5.52i·11-s − 4.79i·13-s + (−3.97 + 1.67i)14-s + (−0.0952 − 3.99i)16-s + 17-s − 1.88i·19-s + (−5.68 − 5.55i)20-s + (3.03 + 7.19i)22-s + 2.95·23-s + ⋯
L(s)  = 1  + (−0.921 + 0.388i)2-s + (0.698 − 0.715i)4-s − 1.77i·5-s + 1.15·7-s + (−0.366 + 0.930i)8-s + (0.689 + 1.63i)10-s − 1.66i·11-s − 1.33i·13-s + (−1.06 + 0.447i)14-s + (−0.0238 − 0.999i)16-s + 0.242·17-s − 0.432i·19-s + (−1.27 − 1.24i)20-s + (0.646 + 1.53i)22-s + 0.616·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.366 + 0.930i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1224 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.366 + 0.930i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1224\)    =    \(2^{3} \cdot 3^{2} \cdot 17\)
Sign: $-0.366 + 0.930i$
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: \((2,\ 1224,\ (\ :1/2),\ -0.366 + 0.930i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.156140378\)
\(L(\frac12)\) \(\approx\) \(1.156140378\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.30 - 0.548i)T \)
3 \( 1 \)
17 \( 1 - T \)
good5 \( 1 + 3.97iT - 5T^{2} \)
7 \( 1 - 3.04T + 7T^{2} \)
11 \( 1 + 5.52iT - 11T^{2} \)
13 \( 1 + 4.79iT - 13T^{2} \)
19 \( 1 + 1.88iT - 19T^{2} \)
23 \( 1 - 2.95T + 23T^{2} \)
29 \( 1 - 8.15iT - 29T^{2} \)
31 \( 1 - 0.314T + 31T^{2} \)
37 \( 1 - 3.47iT - 37T^{2} \)
41 \( 1 - 6.67T + 41T^{2} \)
43 \( 1 + 0.0362iT - 43T^{2} \)
47 \( 1 + 5.44T + 47T^{2} \)
53 \( 1 - 6.18iT - 53T^{2} \)
59 \( 1 - 2.22iT - 59T^{2} \)
61 \( 1 - 8.22iT - 61T^{2} \)
67 \( 1 - 9.92iT - 67T^{2} \)
71 \( 1 - 5.84T + 71T^{2} \)
73 \( 1 - 5.30T + 73T^{2} \)
79 \( 1 + 11.3T + 79T^{2} \)
83 \( 1 + 14.0iT - 83T^{2} \)
89 \( 1 + 10.5T + 89T^{2} \)
97 \( 1 - 3.72T + 97T^{2} \)
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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

−9.066051799977121236531678365095, −8.577176640175170277643047891288, −8.186600400808549149388372397765, −7.37903256952464666458332967967, −5.88261450700795440487623652418, −5.39638598791969017917790347291, −4.67611463438695438985768206449, −3.04242623065013232909199565434, −1.37272046307045913689727966719, −0.72895845405003357304716028319, 1.85489704554639559712522504767, 2.33878817392372017523651888654, 3.68185882355856584619918696184, 4.61240051478577739191875874310, 6.22668762862457734089779926690, 6.97910581587020650951450854358, 7.50455251333977178442441594741, 8.204784087099369998107319129219, 9.544776820968218536922896081922, 9.845015037308118471190491083774

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