Properties

Label 2-152-152.11-c0-0-0
Degree $2$
Conductor $152$
Sign $0.977 - 0.211i$
Analytic cond. $0.0758578$
Root an. cond. $0.275423$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 + 0.866i)2-s + (0.5 − 0.866i)3-s + (−0.499 − 0.866i)4-s + (0.499 + 0.866i)6-s + 0.999·8-s − 11-s − 0.999·12-s + (−0.5 + 0.866i)16-s + (−1 + 1.73i)17-s + (−0.5 − 0.866i)19-s + (0.5 − 0.866i)22-s + (0.5 − 0.866i)24-s + (−0.5 − 0.866i)25-s + 27-s + (−0.499 − 0.866i)32-s + (−0.5 + 0.866i)33-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)2-s + (0.5 − 0.866i)3-s + (−0.499 − 0.866i)4-s + (0.499 + 0.866i)6-s + 0.999·8-s − 11-s − 0.999·12-s + (−0.5 + 0.866i)16-s + (−1 + 1.73i)17-s + (−0.5 − 0.866i)19-s + (0.5 − 0.866i)22-s + (0.5 − 0.866i)24-s + (−0.5 − 0.866i)25-s + 27-s + (−0.499 − 0.866i)32-s + (−0.5 + 0.866i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.977 - 0.211i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.977 - 0.211i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(152\)    =    \(2^{3} \cdot 19\)
Sign: $0.977 - 0.211i$
Analytic conductor: \(0.0758578\)
Root analytic conductor: \(0.275423\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{152} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 152,\ (\ :0),\ 0.977 - 0.211i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5546383861\)
\(L(\frac12)\) \(\approx\) \(0.5546383861\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 - 0.866i)T \)
19 \( 1 + (0.5 + 0.866i)T \)
good3 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
5 \( 1 + (0.5 + 0.866i)T^{2} \)
7 \( 1 - T^{2} \)
11 \( 1 + T + T^{2} \)
13 \( 1 + (0.5 - 0.866i)T^{2} \)
17 \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \)
23 \( 1 + (0.5 - 0.866i)T^{2} \)
29 \( 1 + (0.5 - 0.866i)T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
43 \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \)
47 \( 1 + (0.5 - 0.866i)T^{2} \)
53 \( 1 + (0.5 - 0.866i)T^{2} \)
59 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
61 \( 1 + (0.5 - 0.866i)T^{2} \)
67 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
71 \( 1 + (0.5 + 0.866i)T^{2} \)
73 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
79 \( 1 + (0.5 + 0.866i)T^{2} \)
83 \( 1 + T + T^{2} \)
89 \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \)
97 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
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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

−13.25814109219072467176254385597, −12.81533261395288879606324937369, −10.98704515428475493520638789862, −10.12876269430643243027513528919, −8.657723995035487315844952114077, −8.101073976834170344302537445441, −7.05485255925857981371645894753, −6.06133272168678729103772022551, −4.54406087258176959610073463060, −2.12233152739616025475607167415, 2.56999500303935729279473701049, 3.85397393163801438416186496316, 5.05189013214110699027880545501, 7.18947349529932496515057948897, 8.410114434551979520735328937624, 9.327406285338906020916266976238, 10.08373028466783430772051515063, 10.98317146030726192002502408597, 12.02361452398343865595862925576, 13.15459683331786685532889254614

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