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Properties

Label 2-153-153.113-c1-0-2
Degree $2$
Conductor $153$
Sign $0.00372 - 0.999i$
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  + (0.424 − 0.0558i)2-s + (0.159 + 1.72i)3-s + (−1.75 + 0.470i)4-s + (0.651 + 1.32i)5-s + (0.163 + 0.723i)6-s + (0.282 + 0.139i)7-s + (−1.50 + 0.625i)8-s + (−2.94 + 0.548i)9-s + (0.350 + 0.524i)10-s + (1.64 − 0.558i)11-s + (−1.09 − 2.95i)12-s + (1.19 + 4.45i)13-s + (0.127 + 0.0433i)14-s + (−2.17 + 1.33i)15-s + (2.54 − 1.46i)16-s + (0.980 − 4.00i)17-s + ⋯
L(s)  = 1  + (0.300 − 0.0395i)2-s + (0.0918 + 0.995i)3-s + (−0.877 + 0.235i)4-s + (0.291 + 0.590i)5-s + (0.0669 + 0.295i)6-s + (0.106 + 0.0526i)7-s + (−0.533 + 0.221i)8-s + (−0.983 + 0.182i)9-s + (0.110 + 0.165i)10-s + (0.496 − 0.168i)11-s + (−0.314 − 0.852i)12-s + (0.330 + 1.23i)13-s + (0.0341 + 0.0115i)14-s + (−0.561 + 0.344i)15-s + (0.635 − 0.366i)16-s + (0.237 − 0.971i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.00372 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.00372 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(153\)    =    \(3^{2} \cdot 17\)
Sign: $0.00372 - 0.999i$
Analytic conductor: \(1.22171\)
Root analytic conductor: \(1.10531\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{153} (113, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 153,\ (\ :1/2),\ 0.00372 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.801434 + 0.798454i\)
\(L(\frac12)\) \(\approx\) \(0.801434 + 0.798454i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.159 - 1.72i)T \)
17 \( 1 + (-0.980 + 4.00i)T \)
good2 \( 1 + (-0.424 + 0.0558i)T + (1.93 - 0.517i)T^{2} \)
5 \( 1 + (-0.651 - 1.32i)T + (-3.04 + 3.96i)T^{2} \)
7 \( 1 + (-0.282 - 0.139i)T + (4.26 + 5.55i)T^{2} \)
11 \( 1 + (-1.64 + 0.558i)T + (8.72 - 6.69i)T^{2} \)
13 \( 1 + (-1.19 - 4.45i)T + (-11.2 + 6.5i)T^{2} \)
19 \( 1 + (-2.71 - 1.12i)T + (13.4 + 13.4i)T^{2} \)
23 \( 1 + (-2.37 + 2.70i)T + (-3.00 - 22.8i)T^{2} \)
29 \( 1 + (-0.470 + 7.17i)T + (-28.7 - 3.78i)T^{2} \)
31 \( 1 + (2.94 - 8.68i)T + (-24.5 - 18.8i)T^{2} \)
37 \( 1 + (-0.433 - 2.17i)T + (-34.1 + 14.1i)T^{2} \)
41 \( 1 + (4.44 - 0.291i)T + (40.6 - 5.35i)T^{2} \)
43 \( 1 + (-2.28 - 1.75i)T + (11.1 + 41.5i)T^{2} \)
47 \( 1 + (1.24 - 4.65i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (-4.06 + 9.80i)T + (-37.4 - 37.4i)T^{2} \)
59 \( 1 + (0.857 - 6.51i)T + (-56.9 - 15.2i)T^{2} \)
61 \( 1 + (-1.69 + 3.43i)T + (-37.1 - 48.3i)T^{2} \)
67 \( 1 + (4.04 + 2.33i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (5.89 - 1.17i)T + (65.5 - 27.1i)T^{2} \)
73 \( 1 + (-6.74 - 4.50i)T + (27.9 + 67.4i)T^{2} \)
79 \( 1 + (-2.46 - 7.26i)T + (-62.6 + 48.0i)T^{2} \)
83 \( 1 + (-0.542 - 4.12i)T + (-80.1 + 21.4i)T^{2} \)
89 \( 1 + (-2.40 + 2.40i)T - 89iT^{2} \)
97 \( 1 + (18.6 + 1.22i)T + (96.1 + 12.6i)T^{2} \)
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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

−13.62952606208932333652202628112, −12.07926009942185837273226566499, −11.26999253834474724031046238726, −10.00735501856958290086698452010, −9.260742903467418040426538583146, −8.373281313332956218106371873818, −6.65354188169947328790587096447, −5.26623837693515558402397007925, −4.22318219450361440508743752411, −3.03693007656960759148345810362, 1.20114531781556626570430053320, 3.46987401812739161802859961462, 5.19177235729336725526597914290, 5.99465903528226592539246321288, 7.52145124358680057550556186778, 8.605875768173170152834018132514, 9.388584184336294403180005471540, 10.78425967358001329789539940193, 12.15823475305669355494930938570, 12.98219723561894349701673327777

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