Properties

Label 2-1050-105.104-c1-0-42
Degree $2$
Conductor $1050$
Sign $0.631 + 0.775i$
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  + 2-s + (1.61 − 0.618i)3-s + 4-s + (1.61 − 0.618i)6-s + (−2.61 − 0.381i)7-s + 8-s + (2.23 − 2.00i)9-s − 4.47i·11-s + (1.61 − 0.618i)12-s + 3.23·13-s + (−2.61 − 0.381i)14-s + 16-s + 0.763i·17-s + (2.23 − 2.00i)18-s − 0.472i·19-s + ⋯
L(s)  = 1  + 0.707·2-s + (0.934 − 0.356i)3-s + 0.5·4-s + (0.660 − 0.252i)6-s + (−0.989 − 0.144i)7-s + 0.353·8-s + (0.745 − 0.666i)9-s − 1.34i·11-s + (0.467 − 0.178i)12-s + 0.897·13-s + (−0.699 − 0.102i)14-s + 0.250·16-s + 0.185i·17-s + (0.527 − 0.471i)18-s − 0.108i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.631 + 0.775i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.631 + 0.775i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1050\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $0.631 + 0.775i$
Analytic conductor: \(8.38429\)
Root analytic conductor: \(2.89556\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1050} (1049, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1050,\ (\ :1/2),\ 0.631 + 0.775i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.231609089\)
\(L(\frac12)\) \(\approx\) \(3.231609089\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 + (-1.61 + 0.618i)T \)
5 \( 1 \)
7 \( 1 + (2.61 + 0.381i)T \)
good11 \( 1 + 4.47iT - 11T^{2} \)
13 \( 1 - 3.23T + 13T^{2} \)
17 \( 1 - 0.763iT - 17T^{2} \)
19 \( 1 + 0.472iT - 19T^{2} \)
23 \( 1 - 4T + 23T^{2} \)
29 \( 1 + 5.70iT - 29T^{2} \)
31 \( 1 - 7.23iT - 31T^{2} \)
37 \( 1 + 5.23iT - 37T^{2} \)
41 \( 1 + 6.47T + 41T^{2} \)
43 \( 1 - 12.9iT - 43T^{2} \)
47 \( 1 - 2.47iT - 47T^{2} \)
53 \( 1 - 8.47T + 53T^{2} \)
59 \( 1 + 4.47T + 59T^{2} \)
61 \( 1 + 2.76iT - 61T^{2} \)
67 \( 1 - 12iT - 67T^{2} \)
71 \( 1 + 2.76iT - 71T^{2} \)
73 \( 1 + 6.76T + 73T^{2} \)
79 \( 1 + 8.94T + 79T^{2} \)
83 \( 1 - 16.6iT - 83T^{2} \)
89 \( 1 + 14.4T + 89T^{2} \)
97 \( 1 + 5.23T + 97T^{2} \)
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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

−9.730031967545423111157171622722, −8.821174021007252081804372065324, −8.227673990126708360996837080197, −7.12209344490994265743518999390, −6.41363567199401952692838310842, −5.68204920035240415023245852844, −4.20542923027857181654649450472, −3.36435073271541524964057958057, −2.78253625620746252663217630784, −1.14919653945494743567970880901, 1.82199335561223215722610355937, 2.93725583966169976163329291474, 3.73199676424986754950680523977, 4.60764282010960848294275498108, 5.62441890930335818294982644520, 6.82865924559308352465896084731, 7.29037292717787267910962569560, 8.496729832441635373729530828679, 9.226513701198286483093055861205, 10.04398546168423945366795770767

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