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Properties

Label 2-152-152.27-c1-0-13
Degree $2$
Conductor $152$
Sign $0.402 + 0.915i$
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  + (1.14 − 0.834i)2-s + (−1.05 − 0.606i)3-s + (0.606 − 1.90i)4-s + (2.45 + 1.41i)5-s + (−1.70 + 0.184i)6-s − 0.450i·7-s + (−0.897 − 2.68i)8-s + (−0.763 − 1.32i)9-s + (3.97 − 0.429i)10-s − 2.15·11-s + (−1.79 + 1.63i)12-s + (1.86 + 3.22i)13-s + (−0.376 − 0.514i)14-s + (−1.71 − 2.97i)15-s + (−3.26 − 2.31i)16-s + (−0.716 + 1.24i)17-s + ⋯
L(s)  = 1  + (0.807 − 0.590i)2-s + (−0.606 − 0.350i)3-s + (0.303 − 0.952i)4-s + (1.09 + 0.632i)5-s + (−0.696 + 0.0752i)6-s − 0.170i·7-s + (−0.317 − 0.948i)8-s + (−0.254 − 0.440i)9-s + (1.25 − 0.135i)10-s − 0.651·11-s + (−0.517 + 0.471i)12-s + (0.516 + 0.895i)13-s + (−0.100 − 0.137i)14-s + (−0.443 − 0.767i)15-s + (−0.815 − 0.578i)16-s + (−0.173 + 0.301i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.402 + 0.915i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.402 + 0.915i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(152\)    =    \(2^{3} \cdot 19\)
Sign: $0.402 + 0.915i$
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: \((2,\ 152,\ (\ :1/2),\ 0.402 + 0.915i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.30672 - 0.853290i\)
\(L(\frac12)\) \(\approx\) \(1.30672 - 0.853290i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.14 + 0.834i)T \)
19 \( 1 + (0.0305 - 4.35i)T \)
good3 \( 1 + (1.05 + 0.606i)T + (1.5 + 2.59i)T^{2} \)
5 \( 1 + (-2.45 - 1.41i)T + (2.5 + 4.33i)T^{2} \)
7 \( 1 + 0.450iT - 7T^{2} \)
11 \( 1 + 2.15T + 11T^{2} \)
13 \( 1 + (-1.86 - 3.22i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (0.716 - 1.24i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (-1.12 + 0.652i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-4.22 - 7.32i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 0.497T + 31T^{2} \)
37 \( 1 + 6.72T + 37T^{2} \)
41 \( 1 + (7.30 + 4.21i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2.90 + 5.02i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-0.567 + 0.327i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (3.86 + 6.69i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (12.1 + 6.99i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.13 + 1.23i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.16 + 1.25i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (8.35 - 14.4i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-8.25 + 14.2i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-4.05 + 7.02i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 11.8T + 83T^{2} \)
89 \( 1 + (6.95 - 4.01i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (5.47 + 3.15i)T + (48.5 + 84.0i)T^{2} \)
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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

−12.73168221751295288708139458914, −11.94075362968840413290431473904, −10.80222501285068231088609387581, −10.24924475797263584851759189922, −8.970317043293839796111238353376, −6.86694881440744323807171955919, −6.19379506101887001518413345674, −5.20621488747264355135106664206, −3.43666919629166138341634221442, −1.81839705723372484590095866751, 2.67158364991551557083930519531, 4.73289293067868230197829541798, 5.43850362650757765929005689913, 6.27012710975716018429223533342, 7.85689530462315369604837992685, 8.934228587946402497137413956243, 10.28909866422768409990854211743, 11.28137033306575078089345680898, 12.41823546719022349044123115496, 13.49874104898433333357144531158

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