Properties

Degree $2$
Conductor $1050$
Sign $0.713 + 0.700i$
Motivic weight $1$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + i·2-s + (1.06 − 1.36i)3-s − 4-s + (1.36 + 1.06i)6-s + (2.62 + 0.294i)7-s i·8-s + (−0.716 − 2.91i)9-s − 1.43i·11-s + (−1.06 + 1.36i)12-s − 4.73i·13-s + (−0.294 + 2.62i)14-s + 16-s − 2.59·17-s + (2.91 − 0.716i)18-s − 2.72i·19-s + ⋯
L(s)  = 1  + 0.707i·2-s + (0.616 − 0.786i)3-s − 0.5·4-s + (0.556 + 0.436i)6-s + (0.993 + 0.111i)7-s − 0.353i·8-s + (−0.238 − 0.971i)9-s − 0.431i·11-s + (−0.308 + 0.393i)12-s − 1.31i·13-s + (−0.0786 + 0.702i)14-s + 0.250·16-s − 0.630·17-s + (0.686 − 0.168i)18-s − 0.625i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.713 + 0.700i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.713 + 0.700i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1050\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $0.713 + 0.700i$
Motivic weight: \(1\)
Character: $\chi_{1050} (251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1050,\ (\ :1/2),\ 0.713 + 0.700i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.924820143\)
\(L(\frac12)\) \(\approx\) \(1.924820143\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
3 \( 1 + (-1.06 + 1.36i)T \)
5 \( 1 \)
7 \( 1 + (-2.62 - 0.294i)T \)
good11 \( 1 + 1.43iT - 11T^{2} \)
13 \( 1 + 4.73iT - 13T^{2} \)
17 \( 1 + 2.59T + 17T^{2} \)
19 \( 1 + 2.72iT - 19T^{2} \)
23 \( 1 - 2.82iT - 23T^{2} \)
29 \( 1 - 2.82iT - 29T^{2} \)
31 \( 1 + 5.91iT - 31T^{2} \)
37 \( 1 - 2.39T + 37T^{2} \)
41 \( 1 - 11.1T + 41T^{2} \)
43 \( 1 + 0.432T + 43T^{2} \)
47 \( 1 + 10.3T + 47T^{2} \)
53 \( 1 - 5.69iT - 53T^{2} \)
59 \( 1 - 10.7T + 59T^{2} \)
61 \( 1 + 1.05iT - 61T^{2} \)
67 \( 1 + 9.82T + 67T^{2} \)
71 \( 1 + 13.2iT - 71T^{2} \)
73 \( 1 + 7.58iT - 73T^{2} \)
79 \( 1 - 16.5T + 79T^{2} \)
83 \( 1 - 12.9T + 83T^{2} \)
89 \( 1 - 3.90T + 89T^{2} \)
97 \( 1 + 14.0iT - 97T^{2} \)
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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

−9.403430022753804696517949190009, −8.797464506338472034303421163099, −7.80806303939957560699263966881, −7.70722467298178883522667196942, −6.48050558993119664549762100580, −5.69666853991835714373464226627, −4.72632351752647312949141139447, −3.48053585786902135638185739096, −2.32184513858266542252848332757, −0.847913853050507608860301150241, 1.69784905881433644529121566763, 2.55278126988216747032770406450, 3.96801880132424776936951150896, 4.44729449562709950054066106524, 5.30897641515993745011258508813, 6.72963385196200263540211818029, 7.87133108926971494005766761994, 8.548479872082943499671864207332, 9.293353449106572837827695515651, 9.993570064414559937256400654788

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