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Properties

Label 2-1260-28.27-c1-0-35
Degree $2$
Conductor $1260$
Sign $0.358 - 0.933i$
Analytic cond. $10.0611$
Root an. cond. $3.17193$
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  + (0.449 + 1.34i)2-s + (−1.59 + 1.20i)4-s − i·5-s + (−1.40 − 2.24i)7-s + (−2.33 − 1.59i)8-s + (1.34 − 0.449i)10-s + 3.99i·11-s − 2.50i·13-s + (2.38 − 2.88i)14-s + (1.09 − 3.84i)16-s + 1.07i·17-s + 7.78·19-s + (1.20 + 1.59i)20-s + (−5.35 + 1.79i)22-s + 8.10i·23-s + ⋯
L(s)  = 1  + (0.317 + 0.948i)2-s + (−0.797 + 0.602i)4-s − 0.447i·5-s + (−0.529 − 0.848i)7-s + (−0.825 − 0.564i)8-s + (0.424 − 0.142i)10-s + 1.20i·11-s − 0.694i·13-s + (0.636 − 0.771i)14-s + (0.273 − 0.961i)16-s + 0.261i·17-s + 1.78·19-s + (0.269 + 0.356i)20-s + (−1.14 + 0.382i)22-s + 1.69i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.358 - 0.933i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.358 - 0.933i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1260\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $0.358 - 0.933i$
Analytic conductor: \(10.0611\)
Root analytic conductor: \(3.17193\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1260} (811, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1260,\ (\ :1/2),\ 0.358 - 0.933i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.653715029\)
\(L(\frac12)\) \(\approx\) \(1.653715029\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.449 - 1.34i)T \)
3 \( 1 \)
5 \( 1 + iT \)
7 \( 1 + (1.40 + 2.24i)T \)
good11 \( 1 - 3.99iT - 11T^{2} \)
13 \( 1 + 2.50iT - 13T^{2} \)
17 \( 1 - 1.07iT - 17T^{2} \)
19 \( 1 - 7.78T + 19T^{2} \)
23 \( 1 - 8.10iT - 23T^{2} \)
29 \( 1 - 5.99T + 29T^{2} \)
31 \( 1 - 6.32T + 31T^{2} \)
37 \( 1 - 3.05T + 37T^{2} \)
41 \( 1 + 9.32iT - 41T^{2} \)
43 \( 1 - 8.12iT - 43T^{2} \)
47 \( 1 - 9.01T + 47T^{2} \)
53 \( 1 + 2.86T + 53T^{2} \)
59 \( 1 - 9.68T + 59T^{2} \)
61 \( 1 - 3.21iT - 61T^{2} \)
67 \( 1 + 7.79iT - 67T^{2} \)
71 \( 1 + 5.84iT - 71T^{2} \)
73 \( 1 + 5.73iT - 73T^{2} \)
79 \( 1 + 2.81iT - 79T^{2} \)
83 \( 1 + 12.5T + 83T^{2} \)
89 \( 1 + 3.51iT - 89T^{2} \)
97 \( 1 - 12.0iT - 97T^{2} \)
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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

−9.747821136415432022039355159683, −9.040127540484439732849043963286, −7.76245072602668884312973738774, −7.54269832747415232236254455697, −6.61176732182030530238004814744, −5.61654722109181005919582815098, −4.85525951490022348606682613160, −3.96344258946522134575071738803, −3.02790552339592033439826821286, −0.997558202580968321330242214000, 0.876863490092731376518864127700, 2.59464893076217140072063371514, 3.01478814852718280262367469800, 4.17588410044334244072246221407, 5.25960391909176343687400730436, 6.05357341395165311773769397834, 6.80786459264795170337682357131, 8.287157337933031358681691429920, 8.832782330055963184115767637483, 9.738391425160829025611051138441

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