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Properties

Label 2-1224-1224.1075-c0-0-0
Degree $2$
Conductor $1224$
Sign $0.978 + 0.208i$
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  + (0.866 − 0.5i)2-s + (0.5 + 0.866i)3-s + (0.499 − 0.866i)4-s + (0.866 + 0.499i)6-s − 0.999i·8-s + (−0.499 + 0.866i)9-s + (0.133 − 0.5i)11-s + 0.999·12-s + (−0.5 − 0.866i)16-s + (0.866 − 0.5i)17-s + 0.999i·18-s + i·19-s + (−0.133 − 0.5i)22-s + (0.866 − 0.5i)24-s + (−0.866 + 0.5i)25-s + ⋯
L(s)  = 1  + (0.866 − 0.5i)2-s + (0.5 + 0.866i)3-s + (0.499 − 0.866i)4-s + (0.866 + 0.499i)6-s − 0.999i·8-s + (−0.499 + 0.866i)9-s + (0.133 − 0.5i)11-s + 0.999·12-s + (−0.5 − 0.866i)16-s + (0.866 − 0.5i)17-s + 0.999i·18-s + i·19-s + (−0.133 − 0.5i)22-s + (0.866 − 0.5i)24-s + (−0.866 + 0.5i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.978 + 0.208i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.978 + 0.208i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1224\)    =    \(2^{3} \cdot 3^{2} \cdot 17\)
Sign: $0.978 + 0.208i$
Analytic conductor: \(0.610855\)
Root analytic conductor: \(0.781572\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1224} (1075, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1224,\ (\ :0),\ 0.978 + 0.208i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.980775157\)
\(L(\frac12)\) \(\approx\) \(1.980775157\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 + 0.5i)T \)
3 \( 1 + (-0.5 - 0.866i)T \)
17 \( 1 + (-0.866 + 0.5i)T \)
good5 \( 1 + (0.866 - 0.5i)T^{2} \)
7 \( 1 + (0.866 + 0.5i)T^{2} \)
11 \( 1 + (-0.133 + 0.5i)T + (-0.866 - 0.5i)T^{2} \)
13 \( 1 + (0.5 + 0.866i)T^{2} \)
19 \( 1 - iT - T^{2} \)
23 \( 1 + (-0.866 + 0.5i)T^{2} \)
29 \( 1 + (-0.866 - 0.5i)T^{2} \)
31 \( 1 + (0.866 - 0.5i)T^{2} \)
37 \( 1 + iT^{2} \)
41 \( 1 + (-0.5 + 0.133i)T + (0.866 - 0.5i)T^{2} \)
43 \( 1 + (1.5 - 0.866i)T + (0.5 - 0.866i)T^{2} \)
47 \( 1 + (0.5 - 0.866i)T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 + (1.5 + 0.866i)T + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (0.866 + 0.5i)T^{2} \)
67 \( 1 + (0.866 - 1.5i)T + (-0.5 - 0.866i)T^{2} \)
71 \( 1 + iT^{2} \)
73 \( 1 + (-1.36 + 1.36i)T - iT^{2} \)
79 \( 1 + (0.866 + 0.5i)T^{2} \)
83 \( 1 + (0.5 - 0.866i)T^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 + (0.5 + 0.133i)T + (0.866 + 0.5i)T^{2} \)
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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

−9.883239433072840061465223831240, −9.476059959498292868176680669088, −8.316067289110175295714704784276, −7.50153890296359611364608379162, −6.20928334371560233578238130612, −5.46794358175481620605566442644, −4.64075775107208671185860395387, −3.62578701527683267433455916900, −3.08469312414584096196990082914, −1.75045528332997913670907500389, 1.79440644251576090694828766736, 2.87367022007421255238129452222, 3.79944576769962713264119950802, 4.87763535258130675086256654543, 5.93370211768803933615620164074, 6.61542290625938969288514581710, 7.43749131273058854921173684948, 8.026665177710299822407905608220, 8.854879301062425760447213673620, 9.806735005743090813972579767187

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