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Properties

Label 2-1224-17.2-c1-0-21
Degree $2$
Conductor $1224$
Sign $-0.878 + 0.477i$
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  + (−1.10 − 2.66i)5-s + (1.25 − 3.03i)7-s + (−1.66 − 0.691i)11-s + 1.79i·13-s + (0.985 − 4.00i)17-s + (2.04 + 2.04i)19-s + (−2.97 − 1.23i)23-s + (−2.34 + 2.34i)25-s + (−2.39 − 5.78i)29-s + (−0.769 + 0.318i)31-s − 9.48·35-s + (−3.77 + 1.56i)37-s + (−3.11 + 7.51i)41-s + (−1.19 + 1.19i)43-s + 6.25i·47-s + ⋯
L(s)  = 1  + (−0.493 − 1.19i)5-s + (0.475 − 1.14i)7-s + (−0.503 − 0.208i)11-s + 0.497i·13-s + (0.238 − 0.971i)17-s + (0.468 + 0.468i)19-s + (−0.620 − 0.257i)23-s + (−0.468 + 0.468i)25-s + (−0.444 − 1.07i)29-s + (−0.138 + 0.0572i)31-s − 1.60·35-s + (−0.620 + 0.256i)37-s + (−0.486 + 1.17i)41-s + (−0.182 + 0.182i)43-s + 0.912i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.878 + 0.477i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1224 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.878 + 0.477i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1224\)    =    \(2^{3} \cdot 3^{2} \cdot 17\)
Sign: $-0.878 + 0.477i$
Analytic conductor: \(9.77368\)
Root analytic conductor: \(3.12629\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1224} (937, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1224,\ (\ :1/2),\ -0.878 + 0.477i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.049995007\)
\(L(\frac12)\) \(\approx\) \(1.049995007\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.985 + 4.00i)T \)
good5 \( 1 + (1.10 + 2.66i)T + (-3.53 + 3.53i)T^{2} \)
7 \( 1 + (-1.25 + 3.03i)T + (-4.94 - 4.94i)T^{2} \)
11 \( 1 + (1.66 + 0.691i)T + (7.77 + 7.77i)T^{2} \)
13 \( 1 - 1.79iT - 13T^{2} \)
19 \( 1 + (-2.04 - 2.04i)T + 19iT^{2} \)
23 \( 1 + (2.97 + 1.23i)T + (16.2 + 16.2i)T^{2} \)
29 \( 1 + (2.39 + 5.78i)T + (-20.5 + 20.5i)T^{2} \)
31 \( 1 + (0.769 - 0.318i)T + (21.9 - 21.9i)T^{2} \)
37 \( 1 + (3.77 - 1.56i)T + (26.1 - 26.1i)T^{2} \)
41 \( 1 + (3.11 - 7.51i)T + (-28.9 - 28.9i)T^{2} \)
43 \( 1 + (1.19 - 1.19i)T - 43iT^{2} \)
47 \( 1 - 6.25iT - 47T^{2} \)
53 \( 1 + (4.85 + 4.85i)T + 53iT^{2} \)
59 \( 1 + (4.81 - 4.81i)T - 59iT^{2} \)
61 \( 1 + (-5.35 + 12.9i)T + (-43.1 - 43.1i)T^{2} \)
67 \( 1 + 13.3T + 67T^{2} \)
71 \( 1 + (-9.25 + 3.83i)T + (50.2 - 50.2i)T^{2} \)
73 \( 1 + (-0.877 - 2.11i)T + (-51.6 + 51.6i)T^{2} \)
79 \( 1 + (13.4 + 5.58i)T + (55.8 + 55.8i)T^{2} \)
83 \( 1 + (-8.07 - 8.07i)T + 83iT^{2} \)
89 \( 1 + 16.4iT - 89T^{2} \)
97 \( 1 + (6.34 + 15.3i)T + (-68.5 + 68.5i)T^{2} \)
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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

−9.441468129929827733485195793544, −8.367404127201515643207568530670, −7.86838306133741647885595673442, −7.13565306481382791113935717208, −5.92401736985480213572031786813, −4.81516154621561880019784470502, −4.40696724222079769811523261313, −3.29047963124872771981006873554, −1.61926588041217776757982919474, −0.44506913901164339091540459362, 1.93159167772167607577653588116, 2.93462803799876286652693241000, 3.77755490573068693686716492857, 5.17810490099376225956911351469, 5.78739026141636165205973623702, 6.88291671017115834078319908206, 7.59616550236636423207985091654, 8.379386264799849365961901452300, 9.160982726052205153405372703817, 10.33401590623035804719633126121

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