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Properties

Label 2-152-152.107-c1-0-12
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: $\chi_{152} (107, \cdot )$
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

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

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