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Properties

Label 2-1260-105.74-c0-0-2
Degree $2$
Conductor $1260$
Sign $0.958 - 0.286i$
Analytic cond. $0.628821$
Root an. cond. $0.792982$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.258 + 0.965i)5-s + (0.866 − 0.5i)7-s − i·13-s + (0.707 − 1.22i)17-s + (0.5 + 0.866i)19-s + (0.707 + 1.22i)23-s + (−0.866 − 0.499i)25-s + 1.41i·29-s + (−0.5 + 0.866i)31-s + (0.258 + 0.965i)35-s + (0.866 − 0.5i)37-s + i·43-s + (−0.707 − 1.22i)47-s + (0.499 − 0.866i)49-s + (−1.22 − 0.707i)59-s + ⋯
L(s)  = 1  + (−0.258 + 0.965i)5-s + (0.866 − 0.5i)7-s − i·13-s + (0.707 − 1.22i)17-s + (0.5 + 0.866i)19-s + (0.707 + 1.22i)23-s + (−0.866 − 0.499i)25-s + 1.41i·29-s + (−0.5 + 0.866i)31-s + (0.258 + 0.965i)35-s + (0.866 − 0.5i)37-s + i·43-s + (−0.707 − 1.22i)47-s + (0.499 − 0.866i)49-s + (−1.22 − 0.707i)59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.958 - 0.286i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.958 - 0.286i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1260\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $0.958 - 0.286i$
Analytic conductor: \(0.628821\)
Root analytic conductor: \(0.792982\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1260} (809, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1260,\ (\ :0),\ 0.958 - 0.286i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.145798470\)
\(L(\frac12)\) \(\approx\) \(1.145798470\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 + (0.258 - 0.965i)T \)
7 \( 1 + (-0.866 + 0.5i)T \)
good11 \( 1 + (0.5 + 0.866i)T^{2} \)
13 \( 1 + iT - T^{2} \)
17 \( 1 + (-0.707 + 1.22i)T + (-0.5 - 0.866i)T^{2} \)
19 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
23 \( 1 + (-0.707 - 1.22i)T + (-0.5 + 0.866i)T^{2} \)
29 \( 1 - 1.41iT - T^{2} \)
31 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
37 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 - iT - T^{2} \)
47 \( 1 + (0.707 + 1.22i)T + (-0.5 + 0.866i)T^{2} \)
53 \( 1 + (-0.5 - 0.866i)T^{2} \)
59 \( 1 + (1.22 + 0.707i)T + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
79 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 + (-1.22 + 0.707i)T + (0.5 - 0.866i)T^{2} \)
97 \( 1 - T^{2} \)
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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

−10.08067036318878566400278378009, −9.197620548914343163685202523165, −7.968686745767595465960444894469, −7.56219642881307815875876199509, −6.86283017818192812448711894985, −5.59408866770322562098088525403, −4.96810271385354587704230734143, −3.58943625747657241498378375885, −2.99253390799909931709292648865, −1.40518854163695122598186741464, 1.30642787499293440883789710084, 2.47309353936923648400672123748, 4.04988577145120062114290003017, 4.65767788722302789946416084119, 5.56040995991821587758833302283, 6.44456081033777997372862677182, 7.67678968450708640851697868834, 8.227762220541741701115584361148, 9.017058869701382609666621553651, 9.599218241906931937596057056478

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