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Properties

Label 2-2120-2120.2013-c0-0-1
Degree $2$
Conductor $2120$
Sign $-0.973 + 0.229i$
Analytic cond. $1.05801$
Root an. cond. $1.02859$
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  + (−0.707 + 0.707i)2-s + (1.41 + 1.41i)3-s − 1.00i·4-s + (−0.707 − 0.707i)5-s − 2.00·6-s + (−1 + i)7-s + (0.707 + 0.707i)8-s + 3.00i·9-s + 1.00·10-s + (1.41 − 1.41i)12-s − 1.41i·14-s − 2.00i·15-s − 1.00·16-s + (−1 + i)17-s + (−2.12 − 2.12i)18-s − 1.41i·19-s + ⋯
L(s)  = 1  + (−0.707 + 0.707i)2-s + (1.41 + 1.41i)3-s − 1.00i·4-s + (−0.707 − 0.707i)5-s − 2.00·6-s + (−1 + i)7-s + (0.707 + 0.707i)8-s + 3.00i·9-s + 1.00·10-s + (1.41 − 1.41i)12-s − 1.41i·14-s − 2.00i·15-s − 1.00·16-s + (−1 + i)17-s + (−2.12 − 2.12i)18-s − 1.41i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.973 + 0.229i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.973 + 0.229i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2120\)    =    \(2^{3} \cdot 5 \cdot 53\)
Sign: $-0.973 + 0.229i$
Analytic conductor: \(1.05801\)
Root analytic conductor: \(1.02859\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2120} (2013, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2120,\ (\ :0),\ -0.973 + 0.229i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7952628782\)
\(L(\frac12)\) \(\approx\) \(0.7952628782\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.707 - 0.707i)T \)
5 \( 1 + (0.707 + 0.707i)T \)
53 \( 1 + (-0.707 - 0.707i)T \)
good3 \( 1 + (-1.41 - 1.41i)T + iT^{2} \)
7 \( 1 + (1 - i)T - iT^{2} \)
11 \( 1 - T^{2} \)
13 \( 1 - iT^{2} \)
17 \( 1 + (1 - i)T - iT^{2} \)
19 \( 1 + 1.41iT - T^{2} \)
23 \( 1 - iT^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + iT^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 - iT^{2} \)
47 \( 1 + (1 - i)T - iT^{2} \)
59 \( 1 + T^{2} \)
61 \( 1 - 1.41T + T^{2} \)
67 \( 1 - iT^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - iT^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 + iT^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (-1 + i)T - iT^{2} \)
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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

−9.291854797262328923682252563770, −8.941831486977956458667511683969, −8.487677128194027572894479327810, −7.77249788807370823944423389874, −6.76581529634509184444215719505, −5.62411964070323827492474624280, −4.79456875520641469559023939983, −4.13305858979135606543678104637, −3.04619744821351352488231089371, −2.12836408831529828206713513432, 0.56545436688928726583852173053, 1.94181342104305409998969935241, 2.84728174454495703162073545608, 3.53493539102465213501285410025, 4.04441103460826966468444003425, 6.41394819026063945697705695301, 6.97216199951321295894392588031, 7.35826364912308788768364755015, 8.142750001128152825995822956202, 8.673932609696716651455362840084

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