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Properties

Label 2-2142-17.13-c1-0-22
Degree $2$
Conductor $2142$
Sign $0.999 + 0.0160i$
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 + (−0.480 − 0.480i)5-s + (−0.707 + 0.707i)7-s − i·8-s + (0.480 − 0.480i)10-s + (−1.10 + 1.10i)11-s + 0.639·13-s + (−0.707 − 0.707i)14-s + 16-s + (−4.08 − 0.569i)17-s + 0.314i·19-s + (0.480 + 0.480i)20-s + (−1.10 − 1.10i)22-s + (6.69 − 6.69i)23-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.5·4-s + (−0.214 − 0.214i)5-s + (−0.267 + 0.267i)7-s − 0.353i·8-s + (0.151 − 0.151i)10-s + (−0.334 + 0.334i)11-s + 0.177·13-s + (−0.188 − 0.188i)14-s + 0.250·16-s + (−0.990 − 0.138i)17-s + 0.0721i·19-s + (0.107 + 0.107i)20-s + (−0.236 − 0.236i)22-s + (1.39 − 1.39i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2142 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0160i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2142 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0160i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2142\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 17\)
Sign: $0.999 + 0.0160i$
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: \((2,\ 2142,\ (\ :1/2),\ 0.999 + 0.0160i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.250629871\)
\(L(\frac12)\) \(\approx\) \(1.250629871\)
\(L(\frac{3}{2})\) 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 \( 1 + (4.08 + 0.569i)T \)
good5 \( 1 + (0.480 + 0.480i)T + 5iT^{2} \)
11 \( 1 + (1.10 - 1.10i)T - 11iT^{2} \)
13 \( 1 - 0.639T + 13T^{2} \)
19 \( 1 - 0.314iT - 19T^{2} \)
23 \( 1 + (-6.69 + 6.69i)T - 23iT^{2} \)
29 \( 1 + (1.26 + 1.26i)T + 29iT^{2} \)
31 \( 1 + (-3.32 - 3.32i)T + 31iT^{2} \)
37 \( 1 + (-0.505 - 0.505i)T + 37iT^{2} \)
41 \( 1 + (-0.136 + 0.136i)T - 41iT^{2} \)
43 \( 1 - 8.12iT - 43T^{2} \)
47 \( 1 + 0.181T + 47T^{2} \)
53 \( 1 + 2.80iT - 53T^{2} \)
59 \( 1 + 3.70iT - 59T^{2} \)
61 \( 1 + (-7.57 + 7.57i)T - 61iT^{2} \)
67 \( 1 - 7.60T + 67T^{2} \)
71 \( 1 + (7.52 + 7.52i)T + 71iT^{2} \)
73 \( 1 + (-0.625 - 0.625i)T + 73iT^{2} \)
79 \( 1 + (-11.2 + 11.2i)T - 79iT^{2} \)
83 \( 1 + 11.9iT - 83T^{2} \)
89 \( 1 - 3.81T + 89T^{2} \)
97 \( 1 + (-10.6 - 10.6i)T + 97iT^{2} \)
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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

−8.840853059847511659324057680816, −8.388499775165990535123126696603, −7.50836681265220034750592884008, −6.61855173630660182479557576441, −6.18998026434642427343166434911, −4.88489583657716328087193855767, −4.60600681127939852448864901847, −3.32287871595360253994399330819, −2.25987384128949279712779456282, −0.55640347045015894308801032948, 0.969727431518254309246138601483, 2.28131826024987110933803627555, 3.27487304109395990210563805301, 3.95146155692972265641358924814, 5.00778893789061676694229152702, 5.78160436373824515346894888631, 6.89622861711953758364816716985, 7.49489358494071176176704960142, 8.527563189762936911630821623609, 9.143571531435095750079646282222

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