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Properties

Label 2-147-21.20-c1-0-0
Degree $2$
Conductor $147$
Sign $-0.907 + 0.419i$
Analytic cond. $1.17380$
Root an. cond. $1.08342$
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  + 2.10i·2-s + (−1.13 + 1.30i)3-s − 2.41·4-s − 1.60·5-s + (−2.74 − 2.38i)6-s − 0.870i·8-s + (−0.414 − 2.97i)9-s − 3.37i·10-s + 2.97i·11-s + (2.74 − 3.15i)12-s + 0.317i·13-s + (1.82 − 2.10i)15-s − 2.99·16-s + 3.88·17-s + (6.24 − 0.870i)18-s + 5.22i·19-s + ⋯
L(s)  = 1  + 1.48i·2-s + (−0.656 + 0.754i)3-s − 1.20·4-s − 0.719·5-s + (−1.12 − 0.975i)6-s − 0.307i·8-s + (−0.138 − 0.990i)9-s − 1.06i·10-s + 0.895i·11-s + (0.792 − 0.910i)12-s + 0.0879i·13-s + (0.472 − 0.542i)15-s − 0.749·16-s + 0.941·17-s + (1.47 − 0.205i)18-s + 1.19i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.907 + 0.419i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.907 + 0.419i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(147\)    =    \(3 \cdot 7^{2}\)
Sign: $-0.907 + 0.419i$
Analytic conductor: \(1.17380\)
Root analytic conductor: \(1.08342\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{147} (146, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 147,\ (\ :1/2),\ -0.907 + 0.419i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.148401 - 0.674424i\)
\(L(\frac12)\) \(\approx\) \(0.148401 - 0.674424i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.13 - 1.30i)T \)
7 \( 1 \)
good2 \( 1 - 2.10iT - 2T^{2} \)
5 \( 1 + 1.60T + 5T^{2} \)
11 \( 1 - 2.97iT - 11T^{2} \)
13 \( 1 - 0.317iT - 13T^{2} \)
17 \( 1 - 3.88T + 17T^{2} \)
19 \( 1 - 5.22iT - 19T^{2} \)
23 \( 1 + 1.23iT - 23T^{2} \)
29 \( 1 - 4.71iT - 29T^{2} \)
31 \( 1 - 2.61iT - 31T^{2} \)
37 \( 1 - 9.07T + 37T^{2} \)
41 \( 1 + 6.15T + 41T^{2} \)
43 \( 1 - 2T + 43T^{2} \)
47 \( 1 - 13.2T + 47T^{2} \)
53 \( 1 + 1.74iT - 53T^{2} \)
59 \( 1 - 6.43T + 59T^{2} \)
61 \( 1 + 7.70iT - 61T^{2} \)
67 \( 1 - 2.48T + 67T^{2} \)
71 \( 1 + 4.71iT - 71T^{2} \)
73 \( 1 + 10.3iT - 73T^{2} \)
79 \( 1 + 0.828T + 79T^{2} \)
83 \( 1 - 3.60T + 83T^{2} \)
89 \( 1 + 10.3T + 89T^{2} \)
97 \( 1 + 7.25iT - 97T^{2} \)
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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

−14.19120089691844708489097101548, −12.52500651003821949549821875671, −11.71578978331332446008011809252, −10.44145232429624283406771439238, −9.381365971137434766553906444379, −8.135361291874329589043016370562, −7.20882592285636863191267396531, −6.05191121758066991199647938355, −5.02973151913301200869107537277, −3.90433293438228257629142751242, 0.77338498070463203181121372422, 2.67391096067655859329598825529, 4.14626243564433909347861378325, 5.72756207853992555131417474541, 7.22124737852037633303357433759, 8.368728828773948765938371544100, 9.773620194693128426659457887197, 10.95123487317507977165094015691, 11.50755661072854572913973266899, 12.16909210526343022814400333145

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