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Properties

Label 2-1224-1224.763-c0-0-1
Degree $2$
Conductor $1224$
Sign $0.962 - 0.269i$
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  + (−0.965 + 0.258i)2-s + (0.965 − 0.258i)3-s + (0.866 − 0.499i)4-s + (−0.866 + 0.499i)6-s + (−0.707 + 0.707i)8-s + (0.866 − 0.499i)9-s + (1.20 + 1.57i)11-s + (0.707 − 0.707i)12-s + (0.500 − 0.866i)16-s + (−0.258 + 0.965i)17-s + (−0.707 + 0.707i)18-s + (−1.22 − 1.22i)19-s + (−1.57 − 1.20i)22-s + (−0.500 + 0.866i)24-s + (0.258 + 0.965i)25-s + ⋯
L(s)  = 1  + (−0.965 + 0.258i)2-s + (0.965 − 0.258i)3-s + (0.866 − 0.499i)4-s + (−0.866 + 0.499i)6-s + (−0.707 + 0.707i)8-s + (0.866 − 0.499i)9-s + (1.20 + 1.57i)11-s + (0.707 − 0.707i)12-s + (0.500 − 0.866i)16-s + (−0.258 + 0.965i)17-s + (−0.707 + 0.707i)18-s + (−1.22 − 1.22i)19-s + (−1.57 − 1.20i)22-s + (−0.500 + 0.866i)24-s + (0.258 + 0.965i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.962 - 0.269i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.962 - 0.269i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1224\)    =    \(2^{3} \cdot 3^{2} \cdot 17\)
Sign: $0.962 - 0.269i$
Analytic conductor: \(0.610855\)
Root analytic conductor: \(0.781572\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1224} (763, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1224,\ (\ :0),\ 0.962 - 0.269i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.022908755\)
\(L(\frac12)\) \(\approx\) \(1.022908755\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.965 - 0.258i)T \)
3 \( 1 + (-0.965 + 0.258i)T \)
17 \( 1 + (0.258 - 0.965i)T \)
good5 \( 1 + (-0.258 - 0.965i)T^{2} \)
7 \( 1 + (-0.258 + 0.965i)T^{2} \)
11 \( 1 + (-1.20 - 1.57i)T + (-0.258 + 0.965i)T^{2} \)
13 \( 1 + (-0.5 + 0.866i)T^{2} \)
19 \( 1 + (1.22 + 1.22i)T + iT^{2} \)
23 \( 1 + (-0.965 + 0.258i)T^{2} \)
29 \( 1 + (0.965 + 0.258i)T^{2} \)
31 \( 1 + (0.258 + 0.965i)T^{2} \)
37 \( 1 + (0.707 - 0.707i)T^{2} \)
41 \( 1 + (0.258 + 1.96i)T + (-0.965 + 0.258i)T^{2} \)
43 \( 1 + (-0.133 - 0.5i)T + (-0.866 + 0.5i)T^{2} \)
47 \( 1 + (-0.5 - 0.866i)T^{2} \)
53 \( 1 - iT^{2} \)
59 \( 1 + (1.86 + 0.5i)T + (0.866 + 0.5i)T^{2} \)
61 \( 1 + (-0.258 + 0.965i)T^{2} \)
67 \( 1 + (-0.258 - 0.448i)T + (-0.5 + 0.866i)T^{2} \)
71 \( 1 + (0.707 - 0.707i)T^{2} \)
73 \( 1 + (-1.12 + 0.465i)T + (0.707 - 0.707i)T^{2} \)
79 \( 1 + (0.258 - 0.965i)T^{2} \)
83 \( 1 + (1.36 - 0.366i)T + (0.866 - 0.5i)T^{2} \)
89 \( 1 + 1.41iT - T^{2} \)
97 \( 1 + (0.0340 - 0.258i)T + (-0.965 - 0.258i)T^{2} \)
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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

−9.687596312208311903692843675937, −8.995661126915074034370762455662, −8.585308713221739884853346185534, −7.45157496703378631366275415818, −6.95345304709860687967473469431, −6.27869473136588737199473710244, −4.73736134352897983253940592749, −3.72784478589960943440088554303, −2.30561045301233195540579684365, −1.59640489631176748376900086706, 1.33533570801903698142069368024, 2.61939782861608675002926061176, 3.48156987320893692566985030920, 4.36102571287544915186239041630, 6.07040489820270162450257050368, 6.69903626061674766089109561884, 7.85017804902938332994664479919, 8.419085185589343262981840024351, 9.020756568047307356668627369310, 9.683775851626852210707551314806

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