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Properties

Label 2-1050-15.8-c1-0-26
Degree $2$
Conductor $1050$
Sign $0.900 + 0.434i$
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  + (0.707 + 0.707i)2-s + (−1.46 + 0.931i)3-s + 1.00i·4-s + (−1.69 − 0.373i)6-s + (−0.707 + 0.707i)7-s + (−0.707 + 0.707i)8-s + (1.26 − 2.72i)9-s − 6.30i·11-s + (−0.931 − 1.46i)12-s + (0.977 + 0.977i)13-s − 1.00·14-s − 1.00·16-s + (−4.86 − 4.86i)17-s + (2.81 − 1.02i)18-s − 0.285i·19-s + ⋯
L(s)  = 1  + (0.499 + 0.499i)2-s + (−0.843 + 0.537i)3-s + 0.500i·4-s + (−0.690 − 0.152i)6-s + (−0.267 + 0.267i)7-s + (−0.250 + 0.250i)8-s + (0.421 − 0.906i)9-s − 1.90i·11-s + (−0.268 − 0.421i)12-s + (0.271 + 0.271i)13-s − 0.267·14-s − 0.250·16-s + (−1.18 − 1.18i)17-s + (0.664 − 0.242i)18-s − 0.0655i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.900 + 0.434i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.900 + 0.434i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1050\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $0.900 + 0.434i$
Analytic conductor: \(8.38429\)
Root analytic conductor: \(2.89556\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1050} (743, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1050,\ (\ :1/2),\ 0.900 + 0.434i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.132361313\)
\(L(\frac12)\) \(\approx\) \(1.132361313\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.707 - 0.707i)T \)
3 \( 1 + (1.46 - 0.931i)T \)
5 \( 1 \)
7 \( 1 + (0.707 - 0.707i)T \)
good11 \( 1 + 6.30iT - 11T^{2} \)
13 \( 1 + (-0.977 - 0.977i)T + 13iT^{2} \)
17 \( 1 + (4.86 + 4.86i)T + 17iT^{2} \)
19 \( 1 + 0.285iT - 19T^{2} \)
23 \( 1 + (-4.26 + 4.26i)T - 23iT^{2} \)
29 \( 1 + 3.84T + 29T^{2} \)
31 \( 1 - 6.64T + 31T^{2} \)
37 \( 1 + (0.317 - 0.317i)T - 37iT^{2} \)
41 \( 1 - 4.55iT - 41T^{2} \)
43 \( 1 + (2.07 + 2.07i)T + 43iT^{2} \)
47 \( 1 + (6.69 + 6.69i)T + 47iT^{2} \)
53 \( 1 + (-3.12 + 3.12i)T - 53iT^{2} \)
59 \( 1 - 13.0T + 59T^{2} \)
61 \( 1 - 1.09T + 61T^{2} \)
67 \( 1 + (-5.63 + 5.63i)T - 67iT^{2} \)
71 \( 1 + 5.42iT - 71T^{2} \)
73 \( 1 + (-3.69 - 3.69i)T + 73iT^{2} \)
79 \( 1 + 4.38iT - 79T^{2} \)
83 \( 1 + (1.52 - 1.52i)T - 83iT^{2} \)
89 \( 1 + 8.96T + 89T^{2} \)
97 \( 1 + (1.50 - 1.50i)T - 97iT^{2} \)
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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

−9.829916473396644224785886934974, −8.906519627157106648573547260878, −8.366654040926687265667194358354, −6.84510772452417792599737347532, −6.45905069515973627476261607548, −5.52045516698086781357099910325, −4.82612413024921342307570729791, −3.76755823562504217973082773206, −2.81731224341935341468366480756, −0.50406427720037678625452491275, 1.39757920981732774624858466586, 2.36552194559870983222299493498, 3.95555841166621494451117432310, 4.70737897281991080459215228021, 5.61048665471157410904855968926, 6.63835539239965973218719673677, 7.12358687678681331654900473552, 8.196293593657796098213631408600, 9.475029282733681523783376709734, 10.18044952356847515471744988875

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