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Properties

Label 2-1368-57.26-c2-0-37
Degree $2$
Conductor $1368$
Sign $-0.876 + 0.482i$
Analytic cond. $37.2753$
Root an. cond. $6.10535$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (7.75 − 4.47i)5-s − 8.60·7-s − 12.8i·11-s + (−11.6 + 20.2i)13-s + (22.1 − 12.7i)17-s + (−18.2 + 5.32i)19-s + (0.190 + 0.110i)23-s + (27.5 − 47.7i)25-s + (−33.8 − 19.5i)29-s + 10.5·31-s + (−66.7 + 38.5i)35-s − 20.5·37-s + (50.6 − 29.2i)41-s + (−24.2 − 41.9i)43-s + (−71.7 − 41.4i)47-s + ⋯
L(s)  = 1  + (1.55 − 0.895i)5-s − 1.22·7-s − 1.16i·11-s + (−0.897 + 1.55i)13-s + (1.30 − 0.750i)17-s + (−0.959 + 0.280i)19-s + (0.00829 + 0.00479i)23-s + (1.10 − 1.91i)25-s + (−1.16 − 0.673i)29-s + 0.339·31-s + (−1.90 + 1.10i)35-s − 0.554·37-s + (1.23 − 0.713i)41-s + (−0.563 − 0.975i)43-s + (−1.52 − 0.881i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.876 + 0.482i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.876 + 0.482i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1368\)    =    \(2^{3} \cdot 3^{2} \cdot 19\)
Sign: $-0.876 + 0.482i$
Analytic conductor: \(37.2753\)
Root analytic conductor: \(6.10535\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1368} (881, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1368,\ (\ :1),\ -0.876 + 0.482i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.134134308\)
\(L(\frac12)\) \(\approx\) \(1.134134308\)
\(L(2)\) 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 \( 1 + (18.2 - 5.32i)T \)
good5 \( 1 + (-7.75 + 4.47i)T + (12.5 - 21.6i)T^{2} \)
7 \( 1 + 8.60T + 49T^{2} \)
11 \( 1 + 12.8iT - 121T^{2} \)
13 \( 1 + (11.6 - 20.2i)T + (-84.5 - 146. i)T^{2} \)
17 \( 1 + (-22.1 + 12.7i)T + (144.5 - 250. i)T^{2} \)
23 \( 1 + (-0.190 - 0.110i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + (33.8 + 19.5i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 - 10.5T + 961T^{2} \)
37 \( 1 + 20.5T + 1.36e3T^{2} \)
41 \( 1 + (-50.6 + 29.2i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (24.2 + 41.9i)T + (-924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (71.7 + 41.4i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (22.9 + 13.2i)T + (1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (-46.2 + 26.6i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (36.4 - 63.1i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (44.6 - 77.3i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + (60.2 - 34.7i)T + (2.52e3 - 4.36e3i)T^{2} \)
73 \( 1 + (12.1 + 21.0i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (24.9 + 43.1i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + 21.4iT - 6.88e3T^{2} \)
89 \( 1 + (-45.6 - 26.3i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + (46.2 + 80.1i)T + (-4.70e3 + 8.14e3i)T^{2} \)
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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

−9.141584573365198489636261237394, −8.631687006470716232518122730252, −7.28510223977246629547361822297, −6.36846321279315169495863763318, −5.81879444849274393855191528107, −5.03520714500933559942246968913, −3.85326423473340711756294067060, −2.66704011542795230211824024324, −1.68249153125609701252193366896, −0.28959253537834436100079276279, 1.64514328941700164446203080159, 2.73430704662190719457680512939, 3.31160929520451050998163095849, 4.88629351438128064292018209095, 5.81516808405757085596910924099, 6.34559883243024753011172204649, 7.16889208444069557998571601326, 7.971520233245814024232879713730, 9.452596153105540156937202365251, 9.727469641599448322600194896475

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