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Properties

Label 2-1224-17.16-c3-0-16
Degree $2$
Conductor $1224$
Sign $0.372 - 0.928i$
Analytic cond. $72.2183$
Root an. cond. $8.49813$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 16.4i·5-s + 34.5i·7-s − 7.42i·11-s − 42.0·13-s + (26.0 − 65.0i)17-s + 59.9·19-s − 49.4i·23-s − 144.·25-s + 259. i·29-s − 92.2i·31-s + 566.·35-s + 207. i·37-s − 176. i·41-s − 19.0·43-s − 80.1·47-s + ⋯
L(s)  = 1  − 1.46i·5-s + 1.86i·7-s − 0.203i·11-s − 0.896·13-s + (0.372 − 0.928i)17-s + 0.723·19-s − 0.448i·23-s − 1.15·25-s + 1.66i·29-s − 0.534i·31-s + 2.73·35-s + 0.922i·37-s − 0.673i·41-s − 0.0676·43-s − 0.248·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.372 - 0.928i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1224 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.372 - 0.928i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1224\)    =    \(2^{3} \cdot 3^{2} \cdot 17\)
Sign: $0.372 - 0.928i$
Analytic conductor: \(72.2183\)
Root analytic conductor: \(8.49813\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{1224} (577, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1224,\ (\ :3/2),\ 0.372 - 0.928i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.416839955\)
\(L(\frac12)\) \(\approx\) \(1.416839955\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
17 \( 1 + (-26.0 + 65.0i)T \)
good5 \( 1 + 16.4iT - 125T^{2} \)
7 \( 1 - 34.5iT - 343T^{2} \)
11 \( 1 + 7.42iT - 1.33e3T^{2} \)
13 \( 1 + 42.0T + 2.19e3T^{2} \)
19 \( 1 - 59.9T + 6.85e3T^{2} \)
23 \( 1 + 49.4iT - 1.21e4T^{2} \)
29 \( 1 - 259. iT - 2.43e4T^{2} \)
31 \( 1 + 92.2iT - 2.97e4T^{2} \)
37 \( 1 - 207. iT - 5.06e4T^{2} \)
41 \( 1 + 176. iT - 6.89e4T^{2} \)
43 \( 1 + 19.0T + 7.95e4T^{2} \)
47 \( 1 + 80.1T + 1.03e5T^{2} \)
53 \( 1 + 319.T + 1.48e5T^{2} \)
59 \( 1 + 11.0T + 2.05e5T^{2} \)
61 \( 1 - 712. iT - 2.26e5T^{2} \)
67 \( 1 - 484.T + 3.00e5T^{2} \)
71 \( 1 - 443. iT - 3.57e5T^{2} \)
73 \( 1 - 337. iT - 3.89e5T^{2} \)
79 \( 1 - 840. iT - 4.93e5T^{2} \)
83 \( 1 + 456.T + 5.71e5T^{2} \)
89 \( 1 - 1.20e3T + 7.04e5T^{2} \)
97 \( 1 + 638. iT - 9.12e5T^{2} \)
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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

−9.311276934968794737386249519953, −8.783738999829593670836962666679, −8.156649829937229417805759765191, −7.09135539055404293586867534058, −5.84081964516756798256140359070, −5.19610465509228245022615260630, −4.75477735535171023730690737679, −3.16424649120747781956641210969, −2.21433915606891676805028794632, −0.998374162382016026384798374033, 0.37863360232224632427328326647, 1.81384800321726783740177026813, 3.12084868486201143830074568822, 3.79947096307179180012324277696, 4.75479679321924159304888383871, 6.10626747339216986922584868544, 6.84484229930173712330510145004, 7.53695143728848176700852797220, 7.926090550668766691802219685719, 9.651439359520051932792329855300

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