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Properties

Label 2-153-17.4-c1-0-5
Degree $2$
Conductor $153$
Sign $-0.859 + 0.511i$
Analytic cond. $1.22171$
Root an. cond. $1.10531$
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  − 2.06i·2-s − 2.27·4-s + (0.245 − 0.245i)5-s + (−3.06 − 3.06i)7-s + 0.558i·8-s + (−0.508 − 0.508i)10-s + (−0.516 − 0.516i)11-s + 5.40·13-s + (−6.33 + 6.33i)14-s − 3.38·16-s + (4.09 + 0.516i)17-s + 3.27i·19-s + (−0.558 + 0.558i)20-s + (−1.06 + 1.06i)22-s + (2.51 + 2.51i)23-s + ⋯
L(s)  = 1  − 1.46i·2-s − 1.13·4-s + (0.109 − 0.109i)5-s + (−1.15 − 1.15i)7-s + 0.197i·8-s + (−0.160 − 0.160i)10-s + (−0.155 − 0.155i)11-s + 1.49·13-s + (−1.69 + 1.69i)14-s − 0.846·16-s + (0.992 + 0.125i)17-s + 0.750i·19-s + (−0.124 + 0.124i)20-s + (−0.227 + 0.227i)22-s + (0.524 + 0.524i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.859 + 0.511i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.859 + 0.511i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(153\)    =    \(3^{2} \cdot 17\)
Sign: $-0.859 + 0.511i$
Analytic conductor: \(1.22171\)
Root analytic conductor: \(1.10531\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{153} (55, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 153,\ (\ :1/2),\ -0.859 + 0.511i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.272404 - 0.989246i\)
\(L(\frac12)\) \(\approx\) \(0.272404 - 0.989246i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
17 \( 1 + (-4.09 - 0.516i)T \)
good2 \( 1 + 2.06iT - 2T^{2} \)
5 \( 1 + (-0.245 + 0.245i)T - 5iT^{2} \)
7 \( 1 + (3.06 + 3.06i)T + 7iT^{2} \)
11 \( 1 + (0.516 + 0.516i)T + 11iT^{2} \)
13 \( 1 - 5.40T + 13T^{2} \)
19 \( 1 - 3.27iT - 19T^{2} \)
23 \( 1 + (-2.51 - 2.51i)T + 23iT^{2} \)
29 \( 1 + (-3.09 + 3.09i)T - 29iT^{2} \)
31 \( 1 + (-2.33 + 2.33i)T - 31iT^{2} \)
37 \( 1 + (-2.03 + 2.03i)T - 37iT^{2} \)
41 \( 1 + (8.37 + 8.37i)T + 41iT^{2} \)
43 \( 1 + 3.74iT - 43T^{2} \)
47 \( 1 - 0.476T + 47T^{2} \)
53 \( 1 - 10.1iT - 53T^{2} \)
59 \( 1 - 5.29iT - 59T^{2} \)
61 \( 1 + (-2.50 - 2.50i)T + 61iT^{2} \)
67 \( 1 + 1.55T + 67T^{2} \)
71 \( 1 + (6.33 - 6.33i)T - 71iT^{2} \)
73 \( 1 + (2.89 - 2.89i)T - 73iT^{2} \)
79 \( 1 + (0.574 + 0.574i)T + 79iT^{2} \)
83 \( 1 + 7.21iT - 83T^{2} \)
89 \( 1 + 9.77T + 89T^{2} \)
97 \( 1 + (1.25 - 1.25i)T - 97iT^{2} \)
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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

−12.54419802668334874779230077888, −11.47750980310043866844801723132, −10.47624921996477350189500992924, −9.973473555815135866089412478871, −8.838515733238703234666230562039, −7.28004710258004744620847257726, −5.93956335637872748525875471680, −3.95305113980978264093933001920, −3.25874030886772859303742726327, −1.14477112855727671832938158368, 3.02999070629052448433390526898, 5.02365064636392694927687413353, 6.18460361980783797279154794973, 6.66444637086170095259796425385, 8.219840476175999224937151504423, 8.901485883057897454211195737894, 10.03214108745068050596559770361, 11.49954943406020996707937880641, 12.70187060090746606354218279435, 13.56314545490329868954921254198

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