Properties

Degree $2$
Conductor $1050$
Sign $0.674 + 0.738i$
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 + (1.68 − 0.403i)3-s − 4-s + (−0.403 − 1.68i)6-s + (2.31 − 1.28i)7-s + i·8-s + (2.67 − 1.35i)9-s + 5.34i·11-s + (−1.68 + 0.403i)12-s + 3.95i·13-s + (−1.28 − 2.31i)14-s + 16-s + 7.32·17-s + (−1.35 − 2.67i)18-s + 0.807i·19-s + ⋯
L(s)  = 1  − 0.707i·2-s + (0.972 − 0.232i)3-s − 0.5·4-s + (−0.164 − 0.687i)6-s + (0.874 − 0.484i)7-s + 0.353i·8-s + (0.891 − 0.453i)9-s + 1.61i·11-s + (−0.486 + 0.116i)12-s + 1.09i·13-s + (−0.342 − 0.618i)14-s + 0.250·16-s + 1.77·17-s + (−0.320 − 0.630i)18-s + 0.185i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.674 + 0.738i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.674 + 0.738i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1050\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $0.674 + 0.738i$
Motivic weight: \(1\)
Character: $\chi_{1050} (251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1050,\ (\ :1/2),\ 0.674 + 0.738i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.479230939\)
\(L(\frac12)\) \(\approx\) \(2.479230939\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.68 + 0.403i)T \)
5 \( 1 \)
7 \( 1 + (-2.31 + 1.28i)T \)
good11 \( 1 - 5.34iT - 11T^{2} \)
13 \( 1 - 3.95iT - 13T^{2} \)
17 \( 1 - 7.32T + 17T^{2} \)
19 \( 1 - 0.807iT - 19T^{2} \)
23 \( 1 - 0.281iT - 23T^{2} \)
29 \( 1 + 0.281iT - 29T^{2} \)
31 \( 1 + 9.07iT - 31T^{2} \)
37 \( 1 + 6.06T + 37T^{2} \)
41 \( 1 + 6.15T + 41T^{2} \)
43 \( 1 + 6.34T + 43T^{2} \)
47 \( 1 + 5.78T + 47T^{2} \)
53 \( 1 - 10.9iT - 53T^{2} \)
59 \( 1 - 4.90T + 59T^{2} \)
61 \( 1 + 13.2iT - 61T^{2} \)
67 \( 1 - 6.71T + 67T^{2} \)
71 \( 1 + 3.36iT - 71T^{2} \)
73 \( 1 + 4.98iT - 73T^{2} \)
79 \( 1 + 3.26T + 79T^{2} \)
83 \( 1 + 1.53T + 83T^{2} \)
89 \( 1 - 4.31T + 89T^{2} \)
97 \( 1 + 15.0iT - 97T^{2} \)
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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

−9.846910534967086048563338029934, −9.165004966162355064594236620114, −8.067059721735621138433598637693, −7.59088857094552893467401557657, −6.70743789183636772486312063133, −5.11064730813353802290589482979, −4.29946165503500097490136215925, −3.49767396170278424720713740292, −2.10152311085867092828817791321, −1.47385492856451772742696085532, 1.27895938125377593109773709556, 3.01037386089505590694687465923, 3.60405106280518398707903847363, 5.16822485452048738766478119970, 5.45924005696963449341361191072, 6.79928853955071751561237024372, 7.84806756799422501746192561168, 8.385661745663975104171255067604, 8.735776950548071745067747235418, 9.964073056582078282598371876762

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