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Properties

Label 2-2142-17.13-c1-0-2
Degree $2$
Conductor $2142$
Sign $-0.848 - 0.529i$
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 + (−2.55 − 2.55i)5-s + (−0.707 + 0.707i)7-s − i·8-s + (2.55 − 2.55i)10-s + (1.87 − 1.87i)11-s − 5.11·13-s + (−0.707 − 0.707i)14-s + 16-s + (3.20 + 2.59i)17-s − 2.83i·19-s + (2.55 + 2.55i)20-s + (1.87 + 1.87i)22-s + (3.03 − 3.03i)23-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.5·4-s + (−1.14 − 1.14i)5-s + (−0.267 + 0.267i)7-s − 0.353i·8-s + (0.808 − 0.808i)10-s + (0.565 − 0.565i)11-s − 1.41·13-s + (−0.188 − 0.188i)14-s + 0.250·16-s + (0.777 + 0.629i)17-s − 0.650i·19-s + (0.571 + 0.571i)20-s + (0.399 + 0.399i)22-s + (0.633 − 0.633i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2142 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.848 - 0.529i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2142 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.848 - 0.529i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2142\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 17\)
Sign: $-0.848 - 0.529i$
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: \((2,\ 2142,\ (\ :1/2),\ -0.848 - 0.529i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3164440055\)
\(L(\frac12)\) \(\approx\) \(0.3164440055\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.20 - 2.59i)T \)
good5 \( 1 + (2.55 + 2.55i)T + 5iT^{2} \)
11 \( 1 + (-1.87 + 1.87i)T - 11iT^{2} \)
13 \( 1 + 5.11T + 13T^{2} \)
19 \( 1 + 2.83iT - 19T^{2} \)
23 \( 1 + (-3.03 + 3.03i)T - 23iT^{2} \)
29 \( 1 + (1.95 + 1.95i)T + 29iT^{2} \)
31 \( 1 + (4.60 + 4.60i)T + 31iT^{2} \)
37 \( 1 + (-0.901 - 0.901i)T + 37iT^{2} \)
41 \( 1 + (0.703 - 0.703i)T - 41iT^{2} \)
43 \( 1 - 3.26iT - 43T^{2} \)
47 \( 1 - 2.00T + 47T^{2} \)
53 \( 1 - 12.2iT - 53T^{2} \)
59 \( 1 - 12.7iT - 59T^{2} \)
61 \( 1 + (7.41 - 7.41i)T - 61iT^{2} \)
67 \( 1 + 9.76T + 67T^{2} \)
71 \( 1 + (-1.38 - 1.38i)T + 71iT^{2} \)
73 \( 1 + (3.41 + 3.41i)T + 73iT^{2} \)
79 \( 1 + (6.39 - 6.39i)T - 79iT^{2} \)
83 \( 1 - 2.20iT - 83T^{2} \)
89 \( 1 + 7.06T + 89T^{2} \)
97 \( 1 + (-12.5 - 12.5i)T + 97iT^{2} \)
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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

−9.012271258587930993261602157783, −8.732486699553605602183925780049, −7.65014054733052479581250661759, −7.41692695175448455455393913936, −6.21017226623780105366294317496, −5.42458614273775617361506787892, −4.57725781716450511444186968997, −3.99735847476866231942746847229, −2.83886507324058715277168381966, −1.04515396126045319480111661764, 0.13321478579101924416392143478, 1.84528601316932121076215454067, 3.10582393079152823275960971725, 3.50400375291643576188204942355, 4.49784629132533224402826505446, 5.37381428819480519721860211621, 6.71810122399297108444711984084, 7.33695007172963818982571168416, 7.73592150345729475893082754669, 8.931089158341309876680832078735

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