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Properties

Label 2-1260-7.2-c1-0-2
Degree $2$
Conductor $1260$
Sign $0.399 - 0.916i$
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  + (0.5 − 0.866i)5-s + (2.35 + 1.20i)7-s + (2.85 + 4.94i)11-s − 6.70·13-s + (1.72 + 2.98i)17-s + (−0.629 + 1.09i)19-s + (−3.98 + 6.89i)23-s + (−0.499 − 0.866i)25-s + 7.18·29-s + (0.5 + 0.866i)31-s + (2.22 − 1.43i)35-s + (4.48 − 7.76i)37-s − 10.2·41-s − 4.44·43-s + (0.740 − 1.28i)47-s + ⋯
L(s)  = 1  + (0.223 − 0.387i)5-s + (0.889 + 0.456i)7-s + (0.860 + 1.49i)11-s − 1.86·13-s + (0.418 + 0.724i)17-s + (−0.144 + 0.250i)19-s + (−0.830 + 1.43i)23-s + (−0.0999 − 0.173i)25-s + 1.33·29-s + (0.0898 + 0.155i)31-s + (0.375 − 0.242i)35-s + (0.737 − 1.27i)37-s − 1.59·41-s − 0.678·43-s + (0.108 − 0.187i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.399 - 0.916i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.399 - 0.916i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1260\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $0.399 - 0.916i$
Analytic conductor: \(10.0611\)
Root analytic conductor: \(3.17193\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1260} (541, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1260,\ (\ :1/2),\ 0.399 - 0.916i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.676295854\)
\(L(\frac12)\) \(\approx\) \(1.676295854\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 + (-0.5 + 0.866i)T \)
7 \( 1 + (-2.35 - 1.20i)T \)
good11 \( 1 + (-2.85 - 4.94i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 6.70T + 13T^{2} \)
17 \( 1 + (-1.72 - 2.98i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.629 - 1.09i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (3.98 - 6.89i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 7.18T + 29T^{2} \)
31 \( 1 + (-0.5 - 0.866i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-4.48 + 7.76i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 10.2T + 41T^{2} \)
43 \( 1 + 4.44T + 43T^{2} \)
47 \( 1 + (-0.740 + 1.28i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-3.44 - 5.97i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (0.146 + 0.253i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.129 + 0.224i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.35 - 5.80i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 11.7T + 71T^{2} \)
73 \( 1 + (0.483 + 0.836i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-5.81 + 10.0i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 14.8T + 83T^{2} \)
89 \( 1 + (6.30 - 10.9i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 13.1T + 97T^{2} \)
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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

−9.784244939638956232114940302900, −9.179183826521375172027284908936, −8.108925549901719811303984352581, −7.50459104331517422246812685313, −6.60162868054321539228085000096, −5.42368955865643229770133390346, −4.81530529140326812234001336916, −3.95325330953352549180744078654, −2.31872018502216356391179474501, −1.58691421962066936039440818475, 0.72059499185396748098177922934, 2.26297750059148839374655435621, 3.25914691018663606359232890874, 4.51183673299178084779615880715, 5.16514854020413340083590137166, 6.38734808474383336683654575794, 6.96123804814409769746453428293, 8.057297047176643684979166291154, 8.532112188252655325703785455522, 9.741486321471058848861328592919

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