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Properties

Label 2-1050-3.2-c2-0-19
Degree $2$
Conductor $1050$
Sign $0.961 + 0.273i$
Analytic cond. $28.6104$
Root an. cond. $5.34887$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.41i·2-s + (0.821 − 2.88i)3-s − 2.00·4-s + (−4.08 − 1.16i)6-s − 2.64·7-s + 2.82i·8-s + (−7.64 − 4.74i)9-s + 18.1i·11-s + (−1.64 + 5.77i)12-s − 22.5·13-s + 3.74i·14-s + 4.00·16-s + 0.457i·17-s + (−6.70 + 10.8i)18-s + 30.1·19-s + ⋯
L(s)  = 1  − 0.707i·2-s + (0.273 − 0.961i)3-s − 0.500·4-s + (−0.680 − 0.193i)6-s − 0.377·7-s + 0.353i·8-s + (−0.849 − 0.526i)9-s + 1.65i·11-s + (−0.136 + 0.480i)12-s − 1.73·13-s + 0.267i·14-s + 0.250·16-s + 0.0268i·17-s + (−0.372 + 0.600i)18-s + 1.58·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.961 + 0.273i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.961 + 0.273i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1050\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $0.961 + 0.273i$
Analytic conductor: \(28.6104\)
Root analytic conductor: \(5.34887\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1050} (701, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1050,\ (\ :1),\ 0.961 + 0.273i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.283096330\)
\(L(\frac12)\) \(\approx\) \(1.283096330\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 1.41iT \)
3 \( 1 + (-0.821 + 2.88i)T \)
5 \( 1 \)
7 \( 1 + 2.64T \)
good11 \( 1 - 18.1iT - 121T^{2} \)
13 \( 1 + 22.5T + 169T^{2} \)
17 \( 1 - 0.457iT - 289T^{2} \)
19 \( 1 - 30.1T + 361T^{2} \)
23 \( 1 + 12.3iT - 529T^{2} \)
29 \( 1 - 4.70iT - 841T^{2} \)
31 \( 1 - 45.2T + 961T^{2} \)
37 \( 1 - 32.9T + 1.36e3T^{2} \)
41 \( 1 - 22.9iT - 1.68e3T^{2} \)
43 \( 1 - 20.2T + 1.84e3T^{2} \)
47 \( 1 - 9.67iT - 2.20e3T^{2} \)
53 \( 1 - 5.97iT - 2.80e3T^{2} \)
59 \( 1 - 112. iT - 3.48e3T^{2} \)
61 \( 1 + 56.3T + 3.72e3T^{2} \)
67 \( 1 - 67.1T + 4.48e3T^{2} \)
71 \( 1 + 20.9iT - 5.04e3T^{2} \)
73 \( 1 + 97.7T + 5.32e3T^{2} \)
79 \( 1 - 41.4T + 6.24e3T^{2} \)
83 \( 1 - 121. iT - 6.88e3T^{2} \)
89 \( 1 - 26.8iT - 7.92e3T^{2} \)
97 \( 1 - 151.T + 9.40e3T^{2} \)
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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

−9.706553779375002014474041953969, −9.110465168562684247162350100335, −7.77988458727229232709816548303, −7.39615602928090972294347522012, −6.47667498036846029070964682199, −5.21418370269992922883345014293, −4.37802996229680813424419828110, −2.89744908167777375434582380131, −2.33023717752457179723584419005, −1.04540767449497491910220477448, 0.44978232769951710181293770380, 2.76914315429404488556098715594, 3.49309649869980709547825907117, 4.69234201120329459460195283763, 5.42018844310077813020022958413, 6.19427422677398845519027228559, 7.42350335188752649211858598305, 8.074852859250884056287504307106, 9.014669876910290782668989434852, 9.657795080449413573592018569284

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