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Properties

Label 2-1050-7.4-c1-0-21
Degree $2$
Conductor $1050$
Sign $0.605 + 0.795i$
Analytic cond. $8.38429$
Root an. cond. $2.89556$
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  + (0.5 + 0.866i)2-s + (0.5 − 0.866i)3-s + (−0.499 + 0.866i)4-s + 0.999·6-s + (−2.5 − 0.866i)7-s − 0.999·8-s + (−0.499 − 0.866i)9-s + (2 − 3.46i)11-s + (0.499 + 0.866i)12-s + 4·13-s + (−0.500 − 2.59i)14-s + (−0.5 − 0.866i)16-s + (−1.5 + 2.59i)17-s + (0.499 − 0.866i)18-s + (−3 − 5.19i)19-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (0.288 − 0.499i)3-s + (−0.249 + 0.433i)4-s + 0.408·6-s + (−0.944 − 0.327i)7-s − 0.353·8-s + (−0.166 − 0.288i)9-s + (0.603 − 1.04i)11-s + (0.144 + 0.249i)12-s + 1.10·13-s + (−0.133 − 0.694i)14-s + (−0.125 − 0.216i)16-s + (−0.363 + 0.630i)17-s + (0.117 − 0.204i)18-s + (−0.688 − 1.19i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1050\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $0.605 + 0.795i$
Analytic conductor: \(8.38429\)
Root analytic conductor: \(2.89556\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1050} (151, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1050,\ (\ :1/2),\ 0.605 + 0.795i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.660673493\)
\(L(\frac12)\) \(\approx\) \(1.660673493\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 - 0.866i)T \)
3 \( 1 + (-0.5 + 0.866i)T \)
5 \( 1 \)
7 \( 1 + (2.5 + 0.866i)T \)
good11 \( 1 + (-2 + 3.46i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 4T + 13T^{2} \)
17 \( 1 + (1.5 - 2.59i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (3 + 5.19i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (3.5 + 6.06i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 4T + 29T^{2} \)
31 \( 1 + (-2.5 + 4.33i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (1 + 1.73i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 7T + 41T^{2} \)
43 \( 1 - 2T + 43T^{2} \)
47 \( 1 + (0.5 + 0.866i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-1 + 1.73i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-7 + 12.1i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (6 + 10.3i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (6 - 10.3i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 9T + 71T^{2} \)
73 \( 1 + (3 - 5.19i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-8.5 - 14.7i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 4T + 83T^{2} \)
89 \( 1 + (-3.5 - 6.06i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 7T + 97T^{2} \)
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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

−9.559759764051861110127683069233, −8.617530226995158245551434302029, −8.319576605369241117188942051794, −7.02389647558746370875847266360, −6.32614360972854453090207017458, −5.99676937004566357556620420972, −4.36246107132561924897415241811, −3.63469731031097482624793299935, −2.55034009698614313194599238537, −0.66015070190360516515504171734, 1.60570822393362871205643876305, 2.85872547013559820690040815149, 3.79764884993598755077774260435, 4.47614740692558688018804823087, 5.75936402153922773320879643468, 6.41163176683544734035480011699, 7.56514754076904702347420389497, 8.780002676284681995349139662872, 9.287047571880463630538903988012, 10.13107181354361460055953823594

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