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Properties

Label 2-1224-17.9-c1-0-14
Degree $2$
Conductor $1224$
Sign $0.892 + 0.451i$
Analytic cond. $9.77368$
Root an. cond. $3.12629$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.35 − 3.26i)5-s + (0.666 + 1.60i)7-s + (3.36 − 1.39i)11-s + 5.09i·13-s + (1.10 + 3.97i)17-s + (2.23 − 2.23i)19-s + (5.29 − 2.19i)23-s + (−5.30 − 5.30i)25-s + (−2.25 + 5.45i)29-s + (3.58 + 1.48i)31-s + 6.15·35-s + (0.0611 + 0.0253i)37-s + (−3.97 − 9.58i)41-s + (1.89 + 1.89i)43-s − 11.4i·47-s + ⋯
L(s)  = 1  + (0.605 − 1.46i)5-s + (0.251 + 0.608i)7-s + (1.01 − 0.420i)11-s + 1.41i·13-s + (0.266 + 0.963i)17-s + (0.512 − 0.512i)19-s + (1.10 − 0.457i)23-s + (−1.06 − 1.06i)25-s + (−0.419 + 1.01i)29-s + (0.644 + 0.267i)31-s + 1.04·35-s + (0.0100 + 0.00416i)37-s + (−0.620 − 1.49i)41-s + (0.289 + 0.289i)43-s − 1.66i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.892 + 0.451i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1224 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.892 + 0.451i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1224\)    =    \(2^{3} \cdot 3^{2} \cdot 17\)
Sign: $0.892 + 0.451i$
Analytic conductor: \(9.77368\)
Root analytic conductor: \(3.12629\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1224} (145, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1224,\ (\ :1/2),\ 0.892 + 0.451i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.073657842\)
\(L(\frac12)\) \(\approx\) \(2.073657842\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.10 - 3.97i)T \)
good5 \( 1 + (-1.35 + 3.26i)T + (-3.53 - 3.53i)T^{2} \)
7 \( 1 + (-0.666 - 1.60i)T + (-4.94 + 4.94i)T^{2} \)
11 \( 1 + (-3.36 + 1.39i)T + (7.77 - 7.77i)T^{2} \)
13 \( 1 - 5.09iT - 13T^{2} \)
19 \( 1 + (-2.23 + 2.23i)T - 19iT^{2} \)
23 \( 1 + (-5.29 + 2.19i)T + (16.2 - 16.2i)T^{2} \)
29 \( 1 + (2.25 - 5.45i)T + (-20.5 - 20.5i)T^{2} \)
31 \( 1 + (-3.58 - 1.48i)T + (21.9 + 21.9i)T^{2} \)
37 \( 1 + (-0.0611 - 0.0253i)T + (26.1 + 26.1i)T^{2} \)
41 \( 1 + (3.97 + 9.58i)T + (-28.9 + 28.9i)T^{2} \)
43 \( 1 + (-1.89 - 1.89i)T + 43iT^{2} \)
47 \( 1 + 11.4iT - 47T^{2} \)
53 \( 1 + (2.59 - 2.59i)T - 53iT^{2} \)
59 \( 1 + (2.93 + 2.93i)T + 59iT^{2} \)
61 \( 1 + (1.53 + 3.69i)T + (-43.1 + 43.1i)T^{2} \)
67 \( 1 + 8.84T + 67T^{2} \)
71 \( 1 + (1.37 + 0.569i)T + (50.2 + 50.2i)T^{2} \)
73 \( 1 + (3.83 - 9.26i)T + (-51.6 - 51.6i)T^{2} \)
79 \( 1 + (-13.1 + 5.44i)T + (55.8 - 55.8i)T^{2} \)
83 \( 1 + (2.07 - 2.07i)T - 83iT^{2} \)
89 \( 1 + 2.32iT - 89T^{2} \)
97 \( 1 + (2.22 - 5.36i)T + (-68.5 - 68.5i)T^{2} \)
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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

−9.227986897390371410370244321908, −8.967327809589323334176818578077, −8.463873697655531110389576632554, −7.08051873195725505913828589555, −6.24230136504917328514397497383, −5.32944233895467335752976256205, −4.67307482797752877352524813383, −3.63195282592006708154928865669, −2.00928312257971940617224437888, −1.16427421841640735609977760857, 1.23032400032903346941640626804, 2.73152174446946201420608810293, 3.38290595757971699215656096957, 4.62603598919339874723225688583, 5.77106957311642162879621299695, 6.47675892592634765113797225485, 7.40016555638113715114252503226, 7.78655219541385471388096239832, 9.261987594558129264576760646133, 9.877339461562861794603390445886

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