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Properties

Label 2-1224-1224.835-c0-0-0
Degree $2$
Conductor $1224$
Sign $0.885 + 0.464i$
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 + (−0.207 + 0.158i)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 + (−0.158 + 0.207i)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 + (−0.207 + 0.158i)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 + (−0.158 + 0.207i)22-s + (−0.500 + 0.866i)24-s + (−0.258 − 0.965i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.885 + 0.464i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.885 + 0.464i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1224\)    =    \(2^{3} \cdot 3^{2} \cdot 17\)
Sign: $0.885 + 0.464i$
Analytic conductor: \(0.610855\)
Root analytic conductor: \(0.781572\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1224} (835, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1224,\ (\ :0),\ 0.885 + 0.464i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.482186411\)
\(L(\frac12)\) \(\approx\) \(1.482186411\)
\(L(1)\) 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 \( 1 + (0.207 - 0.158i)T + (0.258 - 0.965i)T^{2} \)
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 \( 1 + (-0.258 + 0.0340i)T + (0.965 - 0.258i)T^{2} \)
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 \( 1 + (-0.607 - 1.46i)T + (-0.707 + 0.707i)T^{2} \)
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 \( 1 + (1.96 + 0.258i)T + (0.965 + 0.258i)T^{2} \)
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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

−9.935516587607050545479482263862, −9.584684666002235041010994234500, −7.908923771730894727689334289752, −7.16440277905359681807119112144, −6.22569877262251862312937959283, −5.54772232783100791310078811425, −4.79855657948403841790503379873, −3.93051356697694019903558372274, −2.85170679576412416942592034022, −1.32592182899375797073284387251, 1.58180261312317314604441717650, 3.00344207681557720482272355052, 4.11073653184501211892677439553, 5.09738983425863414217753524630, 5.65439968543472250583434204283, 6.52445876923075172051147543462, 7.29914615141506013443149189948, 7.943547770866280218098397195661, 9.196965051006806458530324718330, 10.32795752246282691300618210141

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