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Properties

Label 2-147-21.20-c5-0-17
Degree $2$
Conductor $147$
Sign $0.167 - 0.985i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.167 - 0.985i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.167 - 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(147\)    =    \(3 \cdot 7^{2}\)
Sign: $0.167 - 0.985i$
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: \((2,\ 147,\ (\ :5/2),\ 0.167 - 0.985i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.6879294857\)
\(L(\frac12)\) \(\approx\) \(0.6879294857\)
\(L(\frac{7}{2})\) 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

−12.24179422208119033093484661476, −11.30081378963536023730990117422, −10.72991097294759012260963216919, −8.703483490554173089531347290328, −7.913974710300821341408232437768, −7.00335242237948924084548579266, −5.95807142042363398395153161165, −4.94854701691098621771675199035, −3.57862401236256237558542738650, −0.53822837880034941624375697111, 0.58598604158900365028307087706, 2.21602519055259913775071391182, 4.17098721181277689743466433855, 4.46185194166433467897368939860, 6.69681657563565411241978213861, 7.48487581258445168707875069252, 9.209072858352240880142660870702, 10.27040271152105299597012069405, 11.28116847740344355612937715704, 11.73772183411671427873287945186

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