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Properties

Label 2-2142-17.13-c1-0-3
Degree $2$
Conductor $2142$
Sign $0.938 - 0.344i$
Analytic cond. $17.1039$
Root an. cond. $4.13569$
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  − i·2-s − 4-s + (−2.63 − 2.63i)5-s + (−0.707 + 0.707i)7-s + i·8-s + (−2.63 + 2.63i)10-s + (2.31 − 2.31i)11-s − 5.14·13-s + (0.707 + 0.707i)14-s + 16-s + (−4.01 + 0.934i)17-s + 3.27i·19-s + (2.63 + 2.63i)20-s + (−2.31 − 2.31i)22-s + (0.807 − 0.807i)23-s + ⋯
L(s)  = 1  − 0.707i·2-s − 0.5·4-s + (−1.17 − 1.17i)5-s + (−0.267 + 0.267i)7-s + 0.353i·8-s + (−0.833 + 0.833i)10-s + (0.697 − 0.697i)11-s − 1.42·13-s + (0.188 + 0.188i)14-s + 0.250·16-s + (−0.973 + 0.226i)17-s + 0.750i·19-s + (0.589 + 0.589i)20-s + (−0.493 − 0.493i)22-s + (0.168 − 0.168i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2142 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.938 - 0.344i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2142 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.938 - 0.344i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2142\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 17\)
Sign: $0.938 - 0.344i$
Analytic conductor: \(17.1039\)
Root analytic conductor: \(4.13569\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2142} (1135, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2142,\ (\ :1/2),\ 0.938 - 0.344i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4777170183\)
\(L(\frac12)\) \(\approx\) \(0.4777170183\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + iT \)
3 \( 1 \)
7 \( 1 + (0.707 - 0.707i)T \)
17 \( 1 + (4.01 - 0.934i)T \)
good5 \( 1 + (2.63 + 2.63i)T + 5iT^{2} \)
11 \( 1 + (-2.31 + 2.31i)T - 11iT^{2} \)
13 \( 1 + 5.14T + 13T^{2} \)
19 \( 1 - 3.27iT - 19T^{2} \)
23 \( 1 + (-0.807 + 0.807i)T - 23iT^{2} \)
29 \( 1 + (-1.31 - 1.31i)T + 29iT^{2} \)
31 \( 1 + (4 + 4i)T + 31iT^{2} \)
37 \( 1 + (-3.77 - 3.77i)T + 37iT^{2} \)
41 \( 1 + (-0.950 + 0.950i)T - 41iT^{2} \)
43 \( 1 - 2.24iT - 43T^{2} \)
47 \( 1 - 4.62T + 47T^{2} \)
53 \( 1 + 7.17iT - 53T^{2} \)
59 \( 1 + 5.86iT - 59T^{2} \)
61 \( 1 + (9.15 - 9.15i)T - 61iT^{2} \)
67 \( 1 + 4.88T + 67T^{2} \)
71 \( 1 + (-10.3 - 10.3i)T + 71iT^{2} \)
73 \( 1 + (-9.70 - 9.70i)T + 73iT^{2} \)
79 \( 1 + (-7.27 + 7.27i)T - 79iT^{2} \)
83 \( 1 - 2.13iT - 83T^{2} \)
89 \( 1 + 8.41T + 89T^{2} \)
97 \( 1 + (-10.7 - 10.7i)T + 97iT^{2} \)
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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

−9.158656494129576627846758083347, −8.453506194006740019130439406213, −7.86888013661374720147918657358, −6.89504244103476091069380205153, −5.76240984244223351732921479888, −4.83513278395847406517030918329, −4.20383728820065567751398431701, −3.43767228501336874375045766636, −2.23329387912054759240510963030, −0.892465392689106097668495631757, 0.22039654529195074183272085401, 2.35313403281519265024138900995, 3.34327241636378330012201467931, 4.28756538068594581285890886486, 4.84132543524760960276310259952, 6.21711177500643236739204712333, 7.04317069440826118147449431908, 7.20283497220683116480969121854, 7.935961125489580108555500556312, 9.092787355251971887106423789490

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