↑

Properties

Label 2-2142-17.4-c1-0-1
Degree $2$
Conductor $2142$
Sign $-0.948 + 0.316i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + i·2-s − 4-s + (0.222 − 0.222i)5-s + (−0.707 − 0.707i)7-s − i·8-s + (0.222 + 0.222i)10-s + (−1.72 − 1.72i)11-s − 1.09·13-s + (0.707 − 0.707i)14-s + 16-s + (3.72 − 1.77i)17-s + 2.44i·19-s + (−0.222 + 0.222i)20-s + (1.72 − 1.72i)22-s + (−2.05 − 2.05i)23-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.5·4-s + (0.0993 − 0.0993i)5-s + (−0.267 − 0.267i)7-s − 0.353i·8-s + (0.0702 + 0.0702i)10-s + (−0.521 − 0.521i)11-s − 0.305·13-s + (0.188 − 0.188i)14-s + 0.250·16-s + (0.902 − 0.429i)17-s + 0.560i·19-s + (−0.0496 + 0.0496i)20-s + (0.368 − 0.368i)22-s + (−0.427 − 0.427i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2142 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.948 + 0.316i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2142 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.948 + 0.316i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2142\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 17\)
Sign: $-0.948 + 0.316i$
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: \((2,\ 2142,\ (\ :1/2),\ -0.948 + 0.316i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2582174334\)
\(L(\frac12)\) \(\approx\) \(0.2582174334\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.72 + 1.77i)T \)
good5 \( 1 + (-0.222 + 0.222i)T - 5iT^{2} \)
11 \( 1 + (1.72 + 1.72i)T + 11iT^{2} \)
13 \( 1 + 1.09T + 13T^{2} \)
19 \( 1 - 2.44iT - 19T^{2} \)
23 \( 1 + (2.05 + 2.05i)T + 23iT^{2} \)
29 \( 1 + (2.72 - 2.72i)T - 29iT^{2} \)
31 \( 1 + (4 - 4i)T - 31iT^{2} \)
37 \( 1 + (3.12 - 3.12i)T - 37iT^{2} \)
41 \( 1 + (5.95 + 5.95i)T + 41iT^{2} \)
43 \( 1 - 11.5iT - 43T^{2} \)
47 \( 1 + 3.45T + 47T^{2} \)
53 \( 1 + 12.3iT - 53T^{2} \)
59 \( 1 - 7.54iT - 59T^{2} \)
61 \( 1 + (10.3 + 10.3i)T + 61iT^{2} \)
67 \( 1 - 6.54T + 67T^{2} \)
71 \( 1 + (6.29 - 6.29i)T - 71iT^{2} \)
73 \( 1 + (6.95 - 6.95i)T - 73iT^{2} \)
79 \( 1 + (-1.55 - 1.55i)T + 79iT^{2} \)
83 \( 1 + 0.455iT - 83T^{2} \)
89 \( 1 - 1.34T + 89T^{2} \)
97 \( 1 + (8.30 - 8.30i)T - 97iT^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.470633277034610053583754622858, −8.592277545075767152240276903735, −7.898689281662798816190866863901, −7.20968061295944605231960242835, −6.43460286030992938637125228326, −5.47920628008083983463618751682, −5.03165137736807331294077639011, −3.77100436816413782134491872827, −3.04264254398743246665236395388, −1.47184512759649175035051999195, 0.088700047692839213566935500585, 1.73666434860583052981097063000, 2.61368604928706185863934941279, 3.56313472568266439625327169225, 4.49439208618125568345987105924, 5.41276485602597671533864698273, 6.12533740070991945827829677216, 7.27288935069656827935692392367, 7.904054028705665061201880689947, 8.836223988565552855065691974829

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