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Properties

Label 2-147-21.20-c5-0-39
Degree $2$
Conductor $147$
Sign $-0.145 - 0.989i$
Analytic cond. $23.5764$
Root an. cond. $4.85555$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 7.35i·2-s + (15.5 + 0.174i)3-s − 22.0·4-s + 89.2·5-s + (−1.28 + 114. i)6-s + 73.0i·8-s + (242. + 5.44i)9-s + 656. i·10-s − 623. i·11-s + (−344. − 3.85i)12-s + 535. i·13-s + (1.39e3 + 15.5i)15-s − 1.24e3·16-s + 1.06e3·17-s + (−40.0 + 1.78e3i)18-s − 1.09e3i·19-s + ⋯
L(s)  = 1  + 1.29i·2-s + (0.999 + 0.0111i)3-s − 0.689·4-s + 1.59·5-s + (−0.0145 + 1.29i)6-s + 0.403i·8-s + (0.999 + 0.0223i)9-s + 2.07i·10-s − 1.55i·11-s + (−0.689 − 0.00772i)12-s + 0.879i·13-s + (1.59 + 0.0178i)15-s − 1.21·16-s + 0.894·17-s + (−0.0291 + 1.29i)18-s − 0.694i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.145 - 0.989i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.145 - 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(147\)    =    \(3 \cdot 7^{2}\)
Sign: $-0.145 - 0.989i$
Analytic conductor: \(23.5764\)
Root analytic conductor: \(4.85555\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{147} (146, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 147,\ (\ :5/2),\ -0.145 - 0.989i)\)

Particular Values

\(L(3)\) \(\approx\) \(3.850759530\)
\(L(\frac12)\) \(\approx\) \(3.850759530\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-15.5 - 0.174i)T \)
7 \( 1 \)
good2 \( 1 - 7.35iT - 32T^{2} \)
5 \( 1 - 89.2T + 3.12e3T^{2} \)
11 \( 1 + 623. iT - 1.61e5T^{2} \)
13 \( 1 - 535. iT - 3.71e5T^{2} \)
17 \( 1 - 1.06e3T + 1.41e6T^{2} \)
19 \( 1 + 1.09e3iT - 2.47e6T^{2} \)
23 \( 1 - 812. iT - 6.43e6T^{2} \)
29 \( 1 - 6.06e3iT - 2.05e7T^{2} \)
31 \( 1 + 3.24e3iT - 2.86e7T^{2} \)
37 \( 1 - 5.07e3T + 6.93e7T^{2} \)
41 \( 1 + 6.21e3T + 1.15e8T^{2} \)
43 \( 1 + 1.65e4T + 1.47e8T^{2} \)
47 \( 1 + 2.74e4T + 2.29e8T^{2} \)
53 \( 1 + 1.66e4iT - 4.18e8T^{2} \)
59 \( 1 + 2.86e4T + 7.14e8T^{2} \)
61 \( 1 - 2.00e4iT - 8.44e8T^{2} \)
67 \( 1 + 2.49e4T + 1.35e9T^{2} \)
71 \( 1 - 1.20e4iT - 1.80e9T^{2} \)
73 \( 1 - 4.04e4iT - 2.07e9T^{2} \)
79 \( 1 - 7.09e4T + 3.07e9T^{2} \)
83 \( 1 - 3.63e3T + 3.93e9T^{2} \)
89 \( 1 + 8.22e4T + 5.58e9T^{2} \)
97 \( 1 + 5.46e4iT - 8.58e9T^{2} \)
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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

−13.11450151250500896579968181155, −11.28434254780135286987586081029, −9.907882571766473873435586591884, −9.022637537985971172954193226214, −8.286402664364424957869965358203, −6.96188166001621298787736001122, −6.09874109993037415474514955217, −5.05372257685556211618417094705, −3.07225052928367116419068735540, −1.64856547183502283936036747471, 1.43380342855074722755814716944, 2.17744300884401395627408188451, 3.25634042607472601998573428012, 4.80354058810942585464369577529, 6.45097680356472920330241604140, 7.84388433612736964999425211099, 9.348711597154941874173932011757, 9.992490042731482702809109994078, 10.34264990592633146778924251214, 12.12744611206031506519351634950

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