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Properties

Label 2-1224-1224.43-c0-0-0
Degree $2$
Conductor $1224$
Sign $0.380 - 0.924i$
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 + (−0.207 + 1.57i)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 + (1.57 − 0.207i)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 + (−0.207 + 1.57i)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 + (1.57 − 0.207i)22-s + (−0.500 + 0.866i)24-s + (0.965 − 0.258i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.380 - 0.924i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.380 - 0.924i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1224\)    =    \(2^{3} \cdot 3^{2} \cdot 17\)
Sign: $0.380 - 0.924i$
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: \((2,\ 1224,\ (\ :0),\ 0.380 - 0.924i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7382543643\)
\(L(\frac12)\) \(\approx\) \(0.7382543643\)
\(L(1)\) 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 \( 1 + (0.207 - 1.57i)T + (-0.965 - 0.258i)T^{2} \)
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 \( 1 + (0.965 - 1.25i)T + (-0.258 - 0.965i)T^{2} \)
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 \( 1 + (-0.0999 + 0.241i)T + (-0.707 - 0.707i)T^{2} \)
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 \( 1 + (0.741 + 0.965i)T + (-0.258 + 0.965i)T^{2} \)
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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

−10.18454495250965288385279794946, −9.431723982329812207578602860690, −8.718869050816321248670518460850, −7.999204459676145865937258570083, −6.92584083642656163522103630197, −5.52971672688731303497629586405, −4.44107333096364302248957914961, −4.17942290429329232228647309781, −2.80723659926877249523657651291, −1.97520291257327580283414021810, 0.67607471453593577062369550860, 2.35088250057117237587725967461, 3.64862182360841751621489640131, 4.90787054649075981992207594453, 5.92313032711396198135172201666, 6.53812155986259326579310168260, 7.21721565738839589048413137959, 8.215366754427876298638041762540, 8.748735117874109869107143605982, 9.205908247409185838726913295405

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