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Properties

Label 2-2142-17.4-c1-0-25
Degree $2$
Conductor $2142$
Sign $0.486 - 0.873i$
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 + (2.06 − 2.06i)5-s + (0.707 + 0.707i)7-s − i·8-s + (2.06 + 2.06i)10-s + (4.33 + 4.33i)11-s − 1.50·13-s + (−0.707 + 0.707i)14-s + 16-s + (1.55 − 3.82i)17-s + 6.13i·19-s + (−2.06 + 2.06i)20-s + (−4.33 + 4.33i)22-s + (1.76 + 1.76i)23-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.5·4-s + (0.924 − 0.924i)5-s + (0.267 + 0.267i)7-s − 0.353i·8-s + (0.653 + 0.653i)10-s + (1.30 + 1.30i)11-s − 0.418·13-s + (−0.188 + 0.188i)14-s + 0.250·16-s + (0.376 − 0.926i)17-s + 1.40i·19-s + (−0.462 + 0.462i)20-s + (−0.924 + 0.924i)22-s + (0.367 + 0.367i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2142 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.486 - 0.873i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2142 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.486 - 0.873i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2142\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 17\)
Sign: $0.486 - 0.873i$
Analytic conductor: \(17.1039\)
Root analytic conductor: \(4.13569\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2142} (1891, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2142,\ (\ :1/2),\ 0.486 - 0.873i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.222727899\)
\(L(\frac12)\) \(\approx\) \(2.222727899\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.55 + 3.82i)T \)
good5 \( 1 + (-2.06 + 2.06i)T - 5iT^{2} \)
11 \( 1 + (-4.33 - 4.33i)T + 11iT^{2} \)
13 \( 1 + 1.50T + 13T^{2} \)
19 \( 1 - 6.13iT - 19T^{2} \)
23 \( 1 + (-1.76 - 1.76i)T + 23iT^{2} \)
29 \( 1 + (-3.33 + 3.33i)T - 29iT^{2} \)
31 \( 1 + (4 - 4i)T - 31iT^{2} \)
37 \( 1 + (4.55 - 4.55i)T - 37iT^{2} \)
41 \( 1 + (1.72 + 1.72i)T + 41iT^{2} \)
43 \( 1 + 8.19iT - 43T^{2} \)
47 \( 1 - 8.67T + 47T^{2} \)
53 \( 1 + 7.59iT - 53T^{2} \)
59 \( 1 - 11.6iT - 59T^{2} \)
61 \( 1 + (-3.23 - 3.23i)T + 61iT^{2} \)
67 \( 1 - 2.60T + 67T^{2} \)
71 \( 1 + (5.35 - 5.35i)T - 71iT^{2} \)
73 \( 1 + (-5.30 + 5.30i)T - 73iT^{2} \)
79 \( 1 + (2.13 + 2.13i)T + 79iT^{2} \)
83 \( 1 - 3.64iT - 83T^{2} \)
89 \( 1 - 4.62T + 89T^{2} \)
97 \( 1 + (-8.45 + 8.45i)T - 97iT^{2} \)
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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

−9.036605906853289517829465070070, −8.670468320462976676899968240971, −7.49594295344300937328659892078, −6.95874577476312365698888166140, −5.96730804429604747819982033769, −5.27218105300321992743068905973, −4.66768321680816012900254507394, −3.68989769997229756078232510948, −2.07735320396174183404065778390, −1.22871357761208152801894334992, 0.924803343392434026585890105912, 2.07309852634070391593332621840, 3.02649004132058956469751308940, 3.78075649981461116322637683157, 4.85503391242006302485067071857, 5.90518221147245773930301958371, 6.47899976603271294272227783220, 7.30355155028041046031641481793, 8.455152925839775695418526666181, 9.102594233684869593244985805208

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