Properties

Label 2-1260-28.27-c1-0-29
Degree $2$
Conductor $1260$
Sign $0.894 - 0.447i$
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  + (−1.40 + 0.117i)2-s + (1.97 − 0.329i)4-s i·5-s + (0.776 + 2.52i)7-s + (−2.74 + 0.695i)8-s + (0.117 + 1.40i)10-s − 0.556i·11-s − 0.182i·13-s + (−1.38 − 3.47i)14-s + (3.78 − 1.30i)16-s − 2.39i·17-s + 3.08·19-s + (−0.329 − 1.97i)20-s + (0.0651 + 0.784i)22-s + 3.94i·23-s + ⋯
L(s)  = 1  + (−0.996 + 0.0827i)2-s + (0.986 − 0.164i)4-s − 0.447i·5-s + (0.293 + 0.956i)7-s + (−0.969 + 0.246i)8-s + (0.0370 + 0.445i)10-s − 0.167i·11-s − 0.0505i·13-s + (−0.371 − 0.928i)14-s + (0.945 − 0.325i)16-s − 0.580i·17-s + 0.706·19-s + (−0.0737 − 0.441i)20-s + (0.0138 + 0.167i)22-s + 0.822i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1260\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $0.894 - 0.447i$
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: \((2,\ 1260,\ (\ :1/2),\ 0.894 - 0.447i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.081452600\)
\(L(\frac12)\) \(\approx\) \(1.081452600\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.40 - 0.117i)T \)
3 \( 1 \)
5 \( 1 + iT \)
7 \( 1 + (-0.776 - 2.52i)T \)
good11 \( 1 + 0.556iT - 11T^{2} \)
13 \( 1 + 0.182iT - 13T^{2} \)
17 \( 1 + 2.39iT - 17T^{2} \)
19 \( 1 - 3.08T + 19T^{2} \)
23 \( 1 - 3.94iT - 23T^{2} \)
29 \( 1 - 3.20T + 29T^{2} \)
31 \( 1 - 5.68T + 31T^{2} \)
37 \( 1 + 4.98T + 37T^{2} \)
41 \( 1 - 9.64iT - 41T^{2} \)
43 \( 1 + 0.643iT - 43T^{2} \)
47 \( 1 - 3.63T + 47T^{2} \)
53 \( 1 - 6.97T + 53T^{2} \)
59 \( 1 - 8.79T + 59T^{2} \)
61 \( 1 + 14.3iT - 61T^{2} \)
67 \( 1 - 10.0iT - 67T^{2} \)
71 \( 1 - 1.36iT - 71T^{2} \)
73 \( 1 - 10.1iT - 73T^{2} \)
79 \( 1 + 13.0iT - 79T^{2} \)
83 \( 1 - 9.45T + 83T^{2} \)
89 \( 1 + 8.01iT - 89T^{2} \)
97 \( 1 - 0.445iT - 97T^{2} \)
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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

−9.621001398649045797460384008486, −8.905663323815714130488007794277, −8.278922350468482117980787771207, −7.53054905331464068483653779057, −6.54041782583565181925251224660, −5.64287155885665211856315213628, −4.90597382008147369961297409414, −3.28547018294955777577259380849, −2.27695159497733190580614770221, −1.03355700306619953292465556282, 0.808221686156497207727263006904, 2.11312879357620007349872880253, 3.27998536814095593112502858659, 4.30119413229303852741211195822, 5.63986741245441372517390617337, 6.71636000321527545873446080703, 7.19263830410856336874542339741, 8.069092676300803276649281854627, 8.742069451403119895339610608262, 9.783303159193803329843435569152

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