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Properties

Label 2-1224-1224.43-c0-0-1
Degree $2$
Conductor $1224$
Sign $0.133 + 0.991i$
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 + (−0.866 − 0.5i)3-s + (−0.866 + 0.499i)4-s + (−0.258 + 0.965i)6-s + (0.707 + 0.707i)8-s + (0.499 + 0.866i)9-s + (−0.0340 + 0.258i)11-s + 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 + (0.258 − 0.0340i)22-s + (−0.258 − 0.965i)24-s + (0.965 − 0.258i)25-s + ⋯
L(s)  = 1  + (−0.258 − 0.965i)2-s + (−0.866 − 0.5i)3-s + (−0.866 + 0.499i)4-s + (−0.258 + 0.965i)6-s + (0.707 + 0.707i)8-s + (0.499 + 0.866i)9-s + (−0.0340 + 0.258i)11-s + 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 + (0.258 − 0.0340i)22-s + (−0.258 − 0.965i)24-s + (0.965 − 0.258i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.133 + 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.133 + 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1224\)    =    \(2^{3} \cdot 3^{2} \cdot 17\)
Sign: $0.133 + 0.991i$
Analytic conductor: \(0.610855\)
Root analytic conductor: \(0.781572\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1224} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1224,\ (\ :0),\ 0.133 + 0.991i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6652603309\)
\(L(\frac12)\) \(\approx\) \(0.6652603309\)
\(L(1)\) 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 \( 1 + (0.866 + 0.5i)T \)
17 \( 1 + (-0.258 - 0.965i)T \)
good5 \( 1 + (-0.965 + 0.258i)T^{2} \)
7 \( 1 + (-0.965 - 0.258i)T^{2} \)
11 \( 1 + (0.0340 - 0.258i)T + (-0.965 - 0.258i)T^{2} \)
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 \( 1 + (0.158 - 0.207i)T + (-0.258 - 0.965i)T^{2} \)
43 \( 1 + (0.5 - 0.133i)T + (0.866 - 0.5i)T^{2} \)
47 \( 1 + (-0.5 - 0.866i)T^{2} \)
53 \( 1 + iT^{2} \)
59 \( 1 + (-0.5 + 1.86i)T + (-0.866 - 0.5i)T^{2} \)
61 \( 1 + (-0.965 - 0.258i)T^{2} \)
67 \( 1 + (0.258 + 0.448i)T + (-0.5 + 0.866i)T^{2} \)
71 \( 1 + (-0.707 - 0.707i)T^{2} \)
73 \( 1 + (-0.607 + 1.46i)T + (-0.707 - 0.707i)T^{2} \)
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 \( 1 + (-1.20 - 1.57i)T + (-0.258 + 0.965i)T^{2} \)
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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

−9.939937121963225258203014370961, −9.096950682481933731069345805076, −8.132137330223972393544619166869, −7.36450591142990121729760633925, −6.44867456614916575623494370060, −5.27155856168999742728551003098, −4.66515583313290620229465801456, −3.41551467322052247982345986845, −2.21094680216179537342340585705, −0.999077228290856627203969613894, 1.07396773498078002113915333332, 3.33843568530003383231623374809, 4.37687917740981633462104463911, 5.35619891965697921373490424549, 5.74563001585693616711907379391, 6.85901941143709205338997207300, 7.42599370609520408714177114103, 8.501268058710676427909969212672, 9.337757813825604302381340499285, 9.997650181422803139304343802717

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