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Properties

Label 2-152-152.27-c1-0-9
Degree $2$
Conductor $152$
Sign $-0.669 + 0.742i$
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  + (−1.40 − 0.150i)2-s + (−2.03 − 1.17i)3-s + (1.95 + 0.422i)4-s + (1.50 + 0.869i)5-s + (2.67 + 1.95i)6-s − 2.63i·7-s + (−2.68 − 0.888i)8-s + (1.24 + 2.16i)9-s + (−1.98 − 1.44i)10-s − 3.51·11-s + (−3.47 − 3.15i)12-s + (−3.13 − 5.42i)13-s + (−0.395 + 3.70i)14-s + (−2.03 − 3.53i)15-s + (3.64 + 1.65i)16-s + (−0.535 + 0.926i)17-s + ⋯
L(s)  = 1  + (−0.994 − 0.106i)2-s + (−1.17 − 0.676i)3-s + (0.977 + 0.211i)4-s + (0.673 + 0.388i)5-s + (1.09 + 0.797i)6-s − 0.995i·7-s + (−0.949 − 0.314i)8-s + (0.416 + 0.721i)9-s + (−0.628 − 0.458i)10-s − 1.05·11-s + (−1.00 − 0.909i)12-s + (−0.868 − 1.50i)13-s + (−0.105 + 0.989i)14-s + (−0.526 − 0.911i)15-s + (0.910 + 0.413i)16-s + (−0.129 + 0.224i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.669 + 0.742i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.669 + 0.742i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(152\)    =    \(2^{3} \cdot 19\)
Sign: $-0.669 + 0.742i$
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: \((2,\ 152,\ (\ :1/2),\ -0.669 + 0.742i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.159488 - 0.358766i\)
\(L(\frac12)\) \(\approx\) \(0.159488 - 0.358766i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.40 + 0.150i)T \)
19 \( 1 + (3.79 + 2.13i)T \)
good3 \( 1 + (2.03 + 1.17i)T + (1.5 + 2.59i)T^{2} \)
5 \( 1 + (-1.50 - 0.869i)T + (2.5 + 4.33i)T^{2} \)
7 \( 1 + 2.63iT - 7T^{2} \)
11 \( 1 + 3.51T + 11T^{2} \)
13 \( 1 + (3.13 + 5.42i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (0.535 - 0.926i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (-5.79 + 3.34i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.579 - 1.00i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 3.82T + 31T^{2} \)
37 \( 1 - 3.55T + 37T^{2} \)
41 \( 1 + (4.54 + 2.62i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.164 - 0.284i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (6.68 - 3.85i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-0.526 - 0.912i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-8.45 - 4.87i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-11.2 + 6.51i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-12.3 + 7.14i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-0.554 + 0.960i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (5.98 - 10.3i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (1.02 - 1.76i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 11.0T + 83T^{2} \)
89 \( 1 + (-2.18 + 1.26i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-0.354 - 0.204i)T + (48.5 + 84.0i)T^{2} \)
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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

−12.66593257911755738900846260653, −11.27551056673925487531886988860, −10.50547801466043374323017483208, −10.05846350081633618605853825486, −8.259482233621279269709074831408, −7.21052642814610161992638847233, −6.44616294246887925247660694609, −5.24273398533689552199771973070, −2.62462147807573608515204523924, −0.56959409656755576282069745426, 2.23742134881219737919946836810, 4.95605063204108376093152880730, 5.73428634657069796128603259012, 6.86160849824854706080528969865, 8.462453323928998850454228331966, 9.510069578799789947484419964091, 10.12079898905000657518981930760, 11.31790622224040119048737684627, 11.86957548549253117156697733568, 13.07198150447017181009542834680

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