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Properties

Label 2-1368-19.11-c1-0-21
Degree $2$
Conductor $1368$
Sign $-0.0706 + 0.997i$
Analytic cond. $10.9235$
Root an. cond. $3.30507$
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  + (1.17 − 2.02i)5-s + 3.83·7-s − 3.34·11-s + (−3.08 − 5.34i)13-s + (2.59 − 4.50i)17-s + (−3.01 − 3.14i)19-s + (1.17 + 2.02i)23-s + (−0.244 − 0.423i)25-s + (0.0250 + 0.0434i)29-s − 3.43·31-s + (4.48 − 7.77i)35-s − 5.43·37-s + (−3.64 + 6.31i)41-s + (4.43 − 7.67i)43-s + (5.36 + 9.29i)47-s + ⋯
L(s)  = 1  + (0.523 − 0.907i)5-s + 1.44·7-s − 1.00·11-s + (−0.856 − 1.48i)13-s + (0.630 − 1.09i)17-s + (−0.691 − 0.722i)19-s + (0.244 + 0.423i)23-s + (−0.0489 − 0.0847i)25-s + (0.00466 + 0.00807i)29-s − 0.617·31-s + (0.758 − 1.31i)35-s − 0.894·37-s + (−0.569 + 0.986i)41-s + (0.675 − 1.17i)43-s + (0.783 + 1.35i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0706 + 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0706 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1368\)    =    \(2^{3} \cdot 3^{2} \cdot 19\)
Sign: $-0.0706 + 0.997i$
Analytic conductor: \(10.9235\)
Root analytic conductor: \(3.30507\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1368} (505, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1368,\ (\ :1/2),\ -0.0706 + 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.768049243\)
\(L(\frac12)\) \(\approx\) \(1.768049243\)
\(L(\frac{3}{2})\) 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 \( 1 \)
19 \( 1 + (3.01 + 3.14i)T \)
good5 \( 1 + (-1.17 + 2.02i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 - 3.83T + 7T^{2} \)
11 \( 1 + 3.34T + 11T^{2} \)
13 \( 1 + (3.08 + 5.34i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-2.59 + 4.50i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (-1.17 - 2.02i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.0250 - 0.0434i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 3.43T + 31T^{2} \)
37 \( 1 + 5.43T + 37T^{2} \)
41 \( 1 + (3.64 - 6.31i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-4.43 + 7.67i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-5.36 - 9.29i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (1.59 + 2.76i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-1.92 + 3.33i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (1.46 + 2.53i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-5.75 - 9.97i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-0.744 + 1.28i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-3.84 + 6.65i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (0.0875 - 0.151i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 7.00T + 83T^{2} \)
89 \( 1 + (6.77 + 11.7i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-1.84 + 3.19i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.322926752732916094296156712901, −8.516894027407653207586536082832, −7.80221116948596548119162575231, −7.24487438343414045483832860180, −5.61290353639836298577848201906, −5.15718929913027068902190534387, −4.69354972769365040341745124881, −3.03770385433276128181221950909, −2.00779292277317825553925750976, −0.70476373114219965700404239774, 1.80035688434853341050601629456, 2.37065771209089470745190805064, 3.83166163377626236117612193457, 4.79986190400950149751128416501, 5.60928960202573095023840800118, 6.58552601846798544056531450799, 7.39176371640419607604660288564, 8.139326637008540573685124227973, 8.905155221680414411130906876382, 10.04329576284798155438554503438

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