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Properties

Label 2-1224-1224.331-c0-0-0
Degree $2$
Conductor $1224$
Sign $0.911 - 0.412i$
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  + (0.258 + 0.965i)2-s + (−0.258 − 0.965i)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 + 0.158i)11-s + (0.707 + 0.707i)12-s + (0.500 − 0.866i)16-s + (0.965 + 0.258i)17-s + (−0.707 − 0.707i)18-s + (1.22 − 1.22i)19-s + (0.158 + 1.20i)22-s + (−0.500 + 0.866i)24-s + (−0.965 + 0.258i)25-s + ⋯
L(s)  = 1  + (0.258 + 0.965i)2-s + (−0.258 − 0.965i)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 + 0.158i)11-s + (0.707 + 0.707i)12-s + (0.500 − 0.866i)16-s + (0.965 + 0.258i)17-s + (−0.707 − 0.707i)18-s + (1.22 − 1.22i)19-s + (0.158 + 1.20i)22-s + (−0.500 + 0.866i)24-s + (−0.965 + 0.258i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.911 - 0.412i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.911 - 0.412i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1224\)    =    \(2^{3} \cdot 3^{2} \cdot 17\)
Sign: $0.911 - 0.412i$
Analytic conductor: \(0.610855\)
Root analytic conductor: \(0.781572\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1224} (331, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1224,\ (\ :0),\ 0.911 - 0.412i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.059176767\)
\(L(\frac12)\) \(\approx\) \(1.059176767\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.258 - 0.965i)T \)
3 \( 1 + (0.258 + 0.965i)T \)
17 \( 1 + (-0.965 - 0.258i)T \)
good5 \( 1 + (0.965 - 0.258i)T^{2} \)
7 \( 1 + (0.965 + 0.258i)T^{2} \)
11 \( 1 + (-1.20 - 0.158i)T + (0.965 + 0.258i)T^{2} \)
13 \( 1 + (-0.5 + 0.866i)T^{2} \)
19 \( 1 + (-1.22 + 1.22i)T - iT^{2} \)
23 \( 1 + (0.258 + 0.965i)T^{2} \)
29 \( 1 + (-0.258 + 0.965i)T^{2} \)
31 \( 1 + (-0.965 + 0.258i)T^{2} \)
37 \( 1 + (0.707 + 0.707i)T^{2} \)
41 \( 1 + (-0.965 - 0.741i)T + (0.258 + 0.965i)T^{2} \)
43 \( 1 + (-1.86 + 0.5i)T + (0.866 - 0.5i)T^{2} \)
47 \( 1 + (-0.5 - 0.866i)T^{2} \)
53 \( 1 + iT^{2} \)
59 \( 1 + (0.133 - 0.5i)T + (-0.866 - 0.5i)T^{2} \)
61 \( 1 + (0.965 + 0.258i)T^{2} \)
67 \( 1 + (0.965 + 1.67i)T + (-0.5 + 0.866i)T^{2} \)
71 \( 1 + (0.707 + 0.707i)T^{2} \)
73 \( 1 + (1.83 + 0.758i)T + (0.707 + 0.707i)T^{2} \)
79 \( 1 + (-0.965 - 0.258i)T^{2} \)
83 \( 1 + (-0.366 - 1.36i)T + (-0.866 + 0.5i)T^{2} \)
89 \( 1 - 1.41iT - T^{2} \)
97 \( 1 + (1.25 - 0.965i)T + (0.258 - 0.965i)T^{2} \)
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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

−9.526293705077069010558410399152, −9.120615660421147480936207586499, −7.959670605897207792145216842347, −7.47025166955420285874432785678, −6.67855634371846870465125446332, −5.94526216928123463177675212173, −5.19233679422473653005162465507, −4.04919649090401265982845675704, −2.91701035474480357953185250192, −1.18231830807133550756794244010, 1.29341746350696053237959046508, 2.93520718686769804224954247384, 3.77646176579393268455697080914, 4.40400776278342584999541919851, 5.63913442958703769281156030194, 5.96778173226452283816290081905, 7.56416142453260072905423102797, 8.617034092769477412091937083376, 9.482813714982416356293994911296, 9.817039262225021723122822747040

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