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Properties

Label 2-153-17.16-c3-0-1
Degree $2$
Conductor $153$
Sign $-0.973 + 0.230i$
Analytic cond. $9.02729$
Root an. cond. $3.00454$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.37·2-s − 2.37·4-s + 19.4i·5-s + 14.9i·7-s + 24.6·8-s − 46.1i·10-s − 31.1i·11-s − 5.21·13-s − 35.5i·14-s − 39.3·16-s + (−68.2 + 16.1i)17-s − 28·19-s − 46.1i·20-s + 73.8i·22-s + 167. i·23-s + ⋯
L(s)  = 1  − 0.838·2-s − 0.296·4-s + 1.73i·5-s + 0.809i·7-s + 1.08·8-s − 1.45i·10-s − 0.853i·11-s − 0.111·13-s − 0.678i·14-s − 0.615·16-s + (−0.973 + 0.230i)17-s − 0.338·19-s − 0.515i·20-s + 0.715i·22-s + 1.51i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.973 + 0.230i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.973 + 0.230i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(153\)    =    \(3^{2} \cdot 17\)
Sign: $-0.973 + 0.230i$
Analytic conductor: \(9.02729\)
Root analytic conductor: \(3.00454\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{153} (118, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 153,\ (\ :3/2),\ -0.973 + 0.230i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.0413760 - 0.354804i\)
\(L(\frac12)\) \(\approx\) \(0.0413760 - 0.354804i\)
\(L(\frac{5}{2})\) 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 \( 1 + (68.2 - 16.1i)T \)
good2 \( 1 + 2.37T + 8T^{2} \)
5 \( 1 - 19.4iT - 125T^{2} \)
7 \( 1 - 14.9iT - 343T^{2} \)
11 \( 1 + 31.1iT - 1.33e3T^{2} \)
13 \( 1 + 5.21T + 2.19e3T^{2} \)
19 \( 1 + 28T + 6.85e3T^{2} \)
23 \( 1 - 167. iT - 1.21e4T^{2} \)
29 \( 1 + 136. iT - 2.43e4T^{2} \)
31 \( 1 + 50.5iT - 2.97e4T^{2} \)
37 \( 1 + 260. iT - 5.06e4T^{2} \)
41 \( 1 + 183. iT - 6.89e4T^{2} \)
43 \( 1 + 348.T + 7.95e4T^{2} \)
47 \( 1 + 318.T + 1.03e5T^{2} \)
53 \( 1 - 408.T + 1.48e5T^{2} \)
59 \( 1 + 108.T + 2.05e5T^{2} \)
61 \( 1 + 123. iT - 2.26e5T^{2} \)
67 \( 1 + 243.T + 3.00e5T^{2} \)
71 \( 1 + 42.7iT - 3.57e5T^{2} \)
73 \( 1 - 875. iT - 3.89e5T^{2} \)
79 \( 1 - 750. iT - 4.93e5T^{2} \)
83 \( 1 - 472.T + 5.71e5T^{2} \)
89 \( 1 + 376.T + 7.04e5T^{2} \)
97 \( 1 + 303. iT - 9.12e5T^{2} \)
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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

−13.26423313095826873083047079341, −11.58979892973324043379452423849, −10.92099642358655182052173927459, −9.982906150926052388964961884381, −8.993606214395344962837258472363, −7.935833123468706982769577155709, −6.85053751080972153279932985597, −5.67553246607702687187242283928, −3.74296005672867699403823510567, −2.26514826226115607138527428483, 0.23531173591316257050857289444, 1.52795377452636749637134796714, 4.39492273784258925965576551481, 4.86030790318103498005912594351, 6.88664604525149959906888406195, 8.160521765160066673864419793912, 8.811344383586238117211192325397, 9.728151724341153139308217211634, 10.62106022159661130512385059995, 12.10048700137878077581324550878

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