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Properties

Label 2-152-152.27-c1-0-4
Degree $2$
Conductor $152$
Sign $0.0800 - 0.996i$
Analytic cond. $1.21372$
Root an. cond. $1.10169$
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  + (−0.861 + 1.12i)2-s + (2.07 + 1.19i)3-s + (−0.515 − 1.93i)4-s + (0.418 + 0.241i)5-s + (−3.13 + 1.29i)6-s + 1.56i·7-s + (2.61 + 1.08i)8-s + (1.37 + 2.37i)9-s + (−0.631 + 0.261i)10-s + 1.24·11-s + (1.24 − 4.62i)12-s + (−1.80 − 3.11i)13-s + (−1.75 − 1.35i)14-s + (0.579 + 1.00i)15-s + (−3.46 + 1.99i)16-s + (−1.35 + 2.34i)17-s + ⋯
L(s)  = 1  + (−0.609 + 0.792i)2-s + (1.19 + 0.692i)3-s + (−0.257 − 0.966i)4-s + (0.187 + 0.108i)5-s + (−1.27 + 0.528i)6-s + 0.592i·7-s + (0.923 + 0.384i)8-s + (0.457 + 0.793i)9-s + (−0.199 + 0.0825i)10-s + 0.374·11-s + (0.360 − 1.33i)12-s + (−0.499 − 0.864i)13-s + (−0.469 − 0.360i)14-s + (0.149 + 0.258i)15-s + (−0.867 + 0.497i)16-s + (−0.328 + 0.569i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0800 - 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0800 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(152\)    =    \(2^{3} \cdot 19\)
Sign: $0.0800 - 0.996i$
Analytic conductor: \(1.21372\)
Root analytic conductor: \(1.10169\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{152} (27, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 152,\ (\ :1/2),\ 0.0800 - 0.996i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.862409 + 0.795897i\)
\(L(\frac12)\) \(\approx\) \(0.862409 + 0.795897i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.861 - 1.12i)T \)
19 \( 1 + (4.25 - 0.936i)T \)
good3 \( 1 + (-2.07 - 1.19i)T + (1.5 + 2.59i)T^{2} \)
5 \( 1 + (-0.418 - 0.241i)T + (2.5 + 4.33i)T^{2} \)
7 \( 1 - 1.56iT - 7T^{2} \)
11 \( 1 - 1.24T + 11T^{2} \)
13 \( 1 + (1.80 + 3.11i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (1.35 - 2.34i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (-5.52 + 3.19i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.695 - 1.20i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 4.86T + 31T^{2} \)
37 \( 1 + 10.9T + 37T^{2} \)
41 \( 1 + (-1.10 - 0.635i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-4.78 + 8.28i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-7.26 + 4.19i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (1.50 + 2.59i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-4.43 - 2.56i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (9.42 - 5.44i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.22 - 1.86i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (6.62 - 11.4i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (0.494 - 0.856i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (5.81 - 10.0i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 12.1T + 83T^{2} \)
89 \( 1 + (11.1 - 6.44i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-2.42 - 1.40i)T + (48.5 + 84.0i)T^{2} \)
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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

−13.68894439085148302380884273041, −12.38111760648545964845646475732, −10.60521884019673539032893846841, −9.985841026901256353091264576403, −8.684806109453806415710188126243, −8.562541161740095735898955845925, −7.04957763348173353474020974324, −5.71838883723701062648017048035, −4.27774189859978822029536305287, −2.48479740555548000941920239961, 1.66499982697303474609631298976, 2.98180212203831774105144511836, 4.40433265452059933415798184697, 6.92134896771958871245833616475, 7.63332616454816294975991908639, 8.885019186022397962574157685166, 9.357864621755484694695757502968, 10.68998847855759838580873508494, 11.77223487936364274189654072536, 12.85489142173347694901376076475

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