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Properties

Label 2-1029-1029.617-c0-0-0
Degree $2$
Conductor $1029$
Sign $-0.118 + 0.992i$
Analytic cond. $0.513537$
Root an. cond. $0.716615$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.981 − 0.191i)3-s + (0.718 − 0.695i)4-s + (−0.997 − 0.0640i)7-s + (0.926 + 0.375i)9-s + (−0.838 + 0.545i)12-s + (−0.243 − 1.50i)13-s + (0.0320 − 0.999i)16-s + 0.569·19-s + (0.967 + 0.253i)21-s + (0.159 − 0.987i)25-s + (−0.838 − 0.545i)27-s + (−0.761 + 0.648i)28-s + (−1.29 − 0.623i)31-s + (0.926 − 0.375i)36-s + (−0.0995 + 1.03i)37-s + ⋯
L(s)  = 1  + (−0.981 − 0.191i)3-s + (0.718 − 0.695i)4-s + (−0.997 − 0.0640i)7-s + (0.926 + 0.375i)9-s + (−0.838 + 0.545i)12-s + (−0.243 − 1.50i)13-s + (0.0320 − 0.999i)16-s + 0.569·19-s + (0.967 + 0.253i)21-s + (0.159 − 0.987i)25-s + (−0.838 − 0.545i)27-s + (−0.761 + 0.648i)28-s + (−1.29 − 0.623i)31-s + (0.926 − 0.375i)36-s + (−0.0995 + 1.03i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1029 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.118 + 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1029 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.118 + 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1029\)    =    \(3 \cdot 7^{3}\)
Sign: $-0.118 + 0.992i$
Analytic conductor: \(0.513537\)
Root analytic conductor: \(0.716615\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1029} (617, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1029,\ (\ :0),\ -0.118 + 0.992i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7229154815\)
\(L(\frac12)\) \(\approx\) \(0.7229154815\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.981 + 0.191i)T \)
7 \( 1 + (0.997 + 0.0640i)T \)
good2 \( 1 + (-0.718 + 0.695i)T^{2} \)
5 \( 1 + (-0.159 + 0.987i)T^{2} \)
11 \( 1 + (0.0960 + 0.995i)T^{2} \)
13 \( 1 + (0.243 + 1.50i)T + (-0.949 + 0.315i)T^{2} \)
17 \( 1 + (0.981 + 0.191i)T^{2} \)
19 \( 1 - 0.569T + T^{2} \)
23 \( 1 + (-0.967 - 0.253i)T^{2} \)
29 \( 1 + (-0.967 + 0.253i)T^{2} \)
31 \( 1 + (1.29 + 0.623i)T + (0.623 + 0.781i)T^{2} \)
37 \( 1 + (0.0995 - 1.03i)T + (-0.981 - 0.191i)T^{2} \)
41 \( 1 + (-0.284 - 0.958i)T^{2} \)
43 \( 1 + (0.544 + 1.22i)T + (-0.672 + 0.740i)T^{2} \)
47 \( 1 + (-0.518 - 0.855i)T^{2} \)
53 \( 1 + (0.761 + 0.648i)T^{2} \)
59 \( 1 + (-0.284 + 0.958i)T^{2} \)
61 \( 1 + (-0.655 - 0.217i)T + (0.801 + 0.598i)T^{2} \)
67 \( 1 + (-1.76 + 0.851i)T + (0.623 - 0.781i)T^{2} \)
71 \( 1 + (-0.967 - 0.253i)T^{2} \)
73 \( 1 + (0.167 - 0.0942i)T + (0.518 - 0.855i)T^{2} \)
79 \( 1 + (-0.422 - 1.85i)T + (-0.900 + 0.433i)T^{2} \)
83 \( 1 + (-0.991 + 0.127i)T^{2} \)
89 \( 1 + (0.345 + 0.938i)T^{2} \)
97 \( 1 + (-1.03 - 0.496i)T + (0.623 + 0.781i)T^{2} \)
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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

−10.17138223990088710939057899947, −9.473367248865751378540231844592, −8.016443393786069536925003754839, −7.13173223353910088826780550016, −6.47844617235565393184651695564, −5.67044228813707695133618084004, −5.09719250914088601314294452835, −3.56728711738775551970888000199, −2.34917672142442273356553028927, −0.75083616348167282958230224897, 1.82913237573604706671081288483, 3.28124767482795185963149230377, 4.09704018713468520798167208463, 5.30693572459245570608648921375, 6.29160678124247600880559599281, 6.93874332555403529928099574819, 7.47606715566314935856009010879, 8.942389741228000336788486286729, 9.545952505486065919805806011943, 10.47826117665426014421707507243

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