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Properties

Label 2-2142-17.13-c1-0-17
Degree $2$
Conductor $2142$
Sign $0.863 - 0.503i$
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 + (2.39 + 2.39i)5-s + (0.707 − 0.707i)7-s + i·8-s + (2.39 − 2.39i)10-s + (−1.39 + 1.39i)11-s − 1.28·13-s + (−0.707 − 0.707i)14-s + 16-s + (3.78 − 1.62i)17-s + 8.42i·19-s + (−2.39 − 2.39i)20-s + (1.39 + 1.39i)22-s + (2.44 − 2.44i)23-s + ⋯
L(s)  = 1  − 0.707i·2-s − 0.5·4-s + (1.06 + 1.06i)5-s + (0.267 − 0.267i)7-s + 0.353i·8-s + (0.755 − 0.755i)10-s + (−0.419 + 0.419i)11-s − 0.356·13-s + (−0.188 − 0.188i)14-s + 0.250·16-s + (0.918 − 0.394i)17-s + 1.93i·19-s + (−0.534 − 0.534i)20-s + (0.296 + 0.296i)22-s + (0.510 − 0.510i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2142 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.863 - 0.503i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2142 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.863 - 0.503i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2142\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 17\)
Sign: $0.863 - 0.503i$
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: \((2,\ 2142,\ (\ :1/2),\ 0.863 - 0.503i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.943056759\)
\(L(\frac12)\) \(\approx\) \(1.943056759\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.78 + 1.62i)T \)
good5 \( 1 + (-2.39 - 2.39i)T + 5iT^{2} \)
11 \( 1 + (1.39 - 1.39i)T - 11iT^{2} \)
13 \( 1 + 1.28T + 13T^{2} \)
19 \( 1 - 8.42iT - 19T^{2} \)
23 \( 1 + (-2.44 + 2.44i)T - 23iT^{2} \)
29 \( 1 + (-2.63 - 2.63i)T + 29iT^{2} \)
31 \( 1 + (4.74 + 4.74i)T + 31iT^{2} \)
37 \( 1 + (1.22 + 1.22i)T + 37iT^{2} \)
41 \( 1 + (1.33 - 1.33i)T - 41iT^{2} \)
43 \( 1 - 12.0iT - 43T^{2} \)
47 \( 1 - 5.10T + 47T^{2} \)
53 \( 1 + 13.8iT - 53T^{2} \)
59 \( 1 - 12.5iT - 59T^{2} \)
61 \( 1 + (7.70 - 7.70i)T - 61iT^{2} \)
67 \( 1 - 15.9T + 67T^{2} \)
71 \( 1 + (-5.22 - 5.22i)T + 71iT^{2} \)
73 \( 1 + (2.40 + 2.40i)T + 73iT^{2} \)
79 \( 1 + (9.16 - 9.16i)T - 79iT^{2} \)
83 \( 1 - 7.05iT - 83T^{2} \)
89 \( 1 + 0.0821T + 89T^{2} \)
97 \( 1 + (-2.30 - 2.30i)T + 97iT^{2} \)
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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

−9.677222052942515684890456627880, −8.387734481035629116649424661819, −7.61284383187616392790817829158, −6.83298764116577401187054760403, −5.84417959152539220603826290851, −5.25265849781783568825827596064, −4.08905598825965008972454323621, −3.10104364906478981357329590104, −2.33049645780787734506641704402, −1.37292544910225749863042812830, 0.71562536597474398204839143597, 1.95507346935023348151649009221, 3.16817293272250030916669037903, 4.55700854712522430568411234984, 5.27684673540535715871294861638, 5.59710712109034627220408002865, 6.63536580212819761072912082671, 7.46557836699082586075328905347, 8.369276764698582742573429017101, 9.011537647723522877304087904237

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