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Properties

Label 2-152-8.5-c1-0-5
Degree $2$
Conductor $152$
Sign $-0.368 - 0.929i$
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  + (0.855 + 1.12i)2-s + 1.91i·3-s + (−0.537 + 1.92i)4-s − 1.51i·5-s + (−2.16 + 1.64i)6-s + 0.580·7-s + (−2.62 + 1.04i)8-s − 0.682·9-s + (1.70 − 1.29i)10-s + 1.31i·11-s + (−3.69 − 1.03i)12-s − 3.89i·13-s + (0.496 + 0.653i)14-s + 2.90·15-s + (−3.42 − 2.06i)16-s − 1.20·17-s + ⋯
L(s)  = 1  + (0.604 + 0.796i)2-s + 1.10i·3-s + (−0.268 + 0.963i)4-s − 0.676i·5-s + (−0.882 + 0.669i)6-s + 0.219·7-s + (−0.929 + 0.368i)8-s − 0.227·9-s + (0.539 − 0.409i)10-s + 0.397i·11-s + (−1.06 − 0.297i)12-s − 1.07i·13-s + (0.132 + 0.174i)14-s + 0.749·15-s + (−0.855 − 0.517i)16-s − 0.291·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.368 - 0.929i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.368 - 0.929i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(152\)    =    \(2^{3} \cdot 19\)
Sign: $-0.368 - 0.929i$
Analytic conductor: \(1.21372\)
Root analytic conductor: \(1.10169\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{152} (77, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 152,\ (\ :1/2),\ -0.368 - 0.929i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.819876 + 1.20718i\)
\(L(\frac12)\) \(\approx\) \(0.819876 + 1.20718i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.855 - 1.12i)T \)
19 \( 1 + iT \)
good3 \( 1 - 1.91iT - 3T^{2} \)
5 \( 1 + 1.51iT - 5T^{2} \)
7 \( 1 - 0.580T + 7T^{2} \)
11 \( 1 - 1.31iT - 11T^{2} \)
13 \( 1 + 3.89iT - 13T^{2} \)
17 \( 1 + 1.20T + 17T^{2} \)
23 \( 1 - 5.85T + 23T^{2} \)
29 \( 1 + 1.29iT - 29T^{2} \)
31 \( 1 - 2.96T + 31T^{2} \)
37 \( 1 - 1.18iT - 37T^{2} \)
41 \( 1 + 9.04T + 41T^{2} \)
43 \( 1 + 8.38iT - 43T^{2} \)
47 \( 1 + 12.8T + 47T^{2} \)
53 \( 1 - 3.07iT - 53T^{2} \)
59 \( 1 + 0.258iT - 59T^{2} \)
61 \( 1 + 14.7iT - 61T^{2} \)
67 \( 1 - 9.54iT - 67T^{2} \)
71 \( 1 + 6.93T + 71T^{2} \)
73 \( 1 - 15.2T + 73T^{2} \)
79 \( 1 + 13.1T + 79T^{2} \)
83 \( 1 - 3.70iT - 83T^{2} \)
89 \( 1 + 8.36T + 89T^{2} \)
97 \( 1 - 17.0T + 97T^{2} \)
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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

−13.25374501341806000599354656880, −12.63418030241473118596297711905, −11.36064670049293941520783317423, −10.13802841636823581645461935280, −9.055250655979522731405699597908, −8.157031532893760402104650310929, −6.81447355956201950353178523102, −5.20412546503842471816839530962, −4.71025330746753895339818483651, −3.32584049335855546582369680780, 1.60403985319325714110510460330, 3.05861702195791530640353808453, 4.71189742193659149692690123252, 6.33129805548508752256126747356, 7.01176865624889350463757666770, 8.558782441290038564401617202960, 9.862598675065378977331527345522, 11.07256084154977814575904898777, 11.69045175924940252783127253942, 12.75915253881800427355772532776

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