Properties

Label 1-152-152.43-r1-0-0
Degree $1$
Conductor $152$
Sign $-0.977 - 0.211i$
Analytic cond. $16.3346$
Root an. cond. $16.3346$
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  + (−0.939 − 0.342i)3-s + (−0.173 − 0.984i)5-s + (0.5 − 0.866i)7-s + (0.766 + 0.642i)9-s + (−0.5 − 0.866i)11-s + (0.939 − 0.342i)13-s + (−0.173 + 0.984i)15-s + (0.766 − 0.642i)17-s + (−0.766 + 0.642i)21-s + (−0.173 + 0.984i)23-s + (−0.939 + 0.342i)25-s + (−0.5 − 0.866i)27-s + (−0.766 − 0.642i)29-s + (0.5 − 0.866i)31-s + (0.173 + 0.984i)33-s + ⋯
L(s)  = 1  + (−0.939 − 0.342i)3-s + (−0.173 − 0.984i)5-s + (0.5 − 0.866i)7-s + (0.766 + 0.642i)9-s + (−0.5 − 0.866i)11-s + (0.939 − 0.342i)13-s + (−0.173 + 0.984i)15-s + (0.766 − 0.642i)17-s + (−0.766 + 0.642i)21-s + (−0.173 + 0.984i)23-s + (−0.939 + 0.342i)25-s + (−0.5 − 0.866i)27-s + (−0.766 − 0.642i)29-s + (0.5 − 0.866i)31-s + (0.173 + 0.984i)33-s + ⋯

Functional equation

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

Invariants

Degree: \(1\)
Conductor: \(152\)    =    \(2^{3} \cdot 19\)
Sign: $-0.977 - 0.211i$
Analytic conductor: \(16.3346\)
Root analytic conductor: \(16.3346\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{152} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 152,\ (1:\ ),\ -0.977 - 0.211i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1016404541 - 0.9484893750i\)
\(L(\frac12)\) \(\approx\) \(0.1016404541 - 0.9484893750i\)
\(L(1)\) \(\approx\) \(0.6721367430 - 0.4222963984i\)
\(L(1)\) \(\approx\) \(0.6721367430 - 0.4222963984i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
19 \( 1 \)
good3 \( 1 + (-0.939 - 0.342i)T \)
5 \( 1 + (-0.173 - 0.984i)T \)
7 \( 1 + (0.5 - 0.866i)T \)
11 \( 1 + (-0.5 - 0.866i)T \)
13 \( 1 + (0.939 - 0.342i)T \)
17 \( 1 + (0.766 - 0.642i)T \)
23 \( 1 + (-0.173 + 0.984i)T \)
29 \( 1 + (-0.766 - 0.642i)T \)
31 \( 1 + (0.5 - 0.866i)T \)
37 \( 1 - T \)
41 \( 1 + (-0.939 - 0.342i)T \)
43 \( 1 + (0.173 + 0.984i)T \)
47 \( 1 + (-0.766 - 0.642i)T \)
53 \( 1 + (-0.173 + 0.984i)T \)
59 \( 1 + (0.766 - 0.642i)T \)
61 \( 1 + (-0.173 + 0.984i)T \)
67 \( 1 + (0.766 + 0.642i)T \)
71 \( 1 + (-0.173 - 0.984i)T \)
73 \( 1 + (-0.939 - 0.342i)T \)
79 \( 1 + (0.939 + 0.342i)T \)
83 \( 1 + (-0.5 + 0.866i)T \)
89 \( 1 + (-0.939 + 0.342i)T \)
97 \( 1 + (0.766 - 0.642i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−28.17157917178677685285110148670, −27.50034668698642548415942450841, −26.316101769281603322004807190266, −25.51530144500325230750057843672, −24.06242906476539201961464954465, −23.210907651369464106279819278339, −22.4207756787263045746647148962, −21.48923508536379065594118605397, −20.67863955924737163856079474776, −18.87920313131559389997135222511, −18.3024074021478390302405613686, −17.44743065959413895048439232279, −16.079726937683383509415242109872, −15.25404802369424555168372521957, −14.395114157360810172973316596609, −12.68499434576016197320646418849, −11.777763570000237485176375019722, −10.809318276523312199735086271730, −10.00071863076335073845441181426, −8.46143483384154094151396407943, −7.0358683812406125165836719895, −6.01969004389261658338456155930, −4.90268531688664449549481582295, −3.498077155031194324369483300582, −1.81089393402381411105565986576, 0.439863178766526273055229799104, 1.40964368304635516115277071917, 3.7595115039368366119613080215, 5.05379187491369278297404149367, 5.89308997298828081297786646408, 7.474945850524891823771159566465, 8.28891022480181636814500362955, 9.91256648381889272250982075979, 11.0985229252006948440301921702, 11.82621980093412424103746256045, 13.18931319123129733476508001495, 13.69610575193750912087492484590, 15.61892221135750168703369641593, 16.472921097949737923337756584950, 17.19843277200271061847465477971, 18.2370642106606979908076403034, 19.29922448533066127193344040416, 20.64759647525656760895917632335, 21.2231219690442378825014954110, 22.708205426765714125253169171843, 23.57985789704304021119536503975, 24.11370573780905634926384296912, 25.119193568815543828237722307520, 26.56038840695296703113689939350, 27.67159536475679689938623755203

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