Properties

Label 2-129-129.89-c1-0-1
Degree $2$
Conductor $129$
Sign $0.754 - 0.656i$
Analytic cond. $1.03007$
Root an. cond. $1.01492$
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  + (−2.04 − 0.983i)2-s + (−1.67 − 0.426i)3-s + (1.95 + 2.45i)4-s + (−1.64 − 0.507i)5-s + (3.00 + 2.52i)6-s + (−1.29 + 0.746i)7-s + (−0.574 − 2.51i)8-s + (2.63 + 1.43i)9-s + (2.86 + 2.65i)10-s + (2.89 + 2.31i)11-s + (−2.23 − 4.95i)12-s + (0.667 − 0.619i)13-s + (3.37 − 0.252i)14-s + (2.54 + 1.55i)15-s + (0.0941 − 0.412i)16-s + (1.50 + 4.86i)17-s + ⋯
L(s)  = 1  + (−1.44 − 0.695i)2-s + (−0.969 − 0.246i)3-s + (0.978 + 1.22i)4-s + (−0.736 − 0.227i)5-s + (1.22 + 1.02i)6-s + (−0.488 + 0.282i)7-s + (−0.203 − 0.890i)8-s + (0.878 + 0.477i)9-s + (0.905 + 0.839i)10-s + (0.873 + 0.696i)11-s + (−0.646 − 1.43i)12-s + (0.185 − 0.171i)13-s + (0.902 − 0.0676i)14-s + (0.657 + 0.401i)15-s + (0.0235 − 0.103i)16-s + (0.364 + 1.18i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 129 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.754 - 0.656i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 129 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.754 - 0.656i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(129\)    =    \(3 \cdot 43\)
Sign: $0.754 - 0.656i$
Analytic conductor: \(1.03007\)
Root analytic conductor: \(1.01492\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{129} (89, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 129,\ (\ :1/2),\ 0.754 - 0.656i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.242983 + 0.0908806i\)
\(L(\frac12)\) \(\approx\) \(0.242983 + 0.0908806i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.67 + 0.426i)T \)
43 \( 1 + (-5.78 - 3.08i)T \)
good2 \( 1 + (2.04 + 0.983i)T + (1.24 + 1.56i)T^{2} \)
5 \( 1 + (1.64 + 0.507i)T + (4.13 + 2.81i)T^{2} \)
7 \( 1 + (1.29 - 0.746i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-2.89 - 2.31i)T + (2.44 + 10.7i)T^{2} \)
13 \( 1 + (-0.667 + 0.619i)T + (0.971 - 12.9i)T^{2} \)
17 \( 1 + (-1.50 - 4.86i)T + (-14.0 + 9.57i)T^{2} \)
19 \( 1 + (-0.686 - 4.55i)T + (-18.1 + 5.60i)T^{2} \)
23 \( 1 + (3.57 - 1.40i)T + (16.8 - 15.6i)T^{2} \)
29 \( 1 + (-0.148 - 1.97i)T + (-28.6 + 4.32i)T^{2} \)
31 \( 1 + (8.90 - 6.07i)T + (11.3 - 28.8i)T^{2} \)
37 \( 1 + (-7.14 - 4.12i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-4.98 + 10.3i)T + (-25.5 - 32.0i)T^{2} \)
47 \( 1 + (6.62 - 5.27i)T + (10.4 - 45.8i)T^{2} \)
53 \( 1 + (-3.64 + 3.92i)T + (-3.96 - 52.8i)T^{2} \)
59 \( 1 + (-7.38 - 1.68i)T + (53.1 + 25.5i)T^{2} \)
61 \( 1 + (1.68 - 2.47i)T + (-22.2 - 56.7i)T^{2} \)
67 \( 1 + (-0.760 + 0.114i)T + (64.0 - 19.7i)T^{2} \)
71 \( 1 + (1.83 - 4.66i)T + (-52.0 - 48.2i)T^{2} \)
73 \( 1 + (2.55 + 2.75i)T + (-5.45 + 72.7i)T^{2} \)
79 \( 1 + (3.40 + 5.89i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (5.20 + 0.390i)T + (82.0 + 12.3i)T^{2} \)
89 \( 1 + (0.845 - 11.2i)T + (-88.0 - 13.2i)T^{2} \)
97 \( 1 + (-2.59 + 3.25i)T + (-21.5 - 94.5i)T^{2} \)
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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

−12.59947559599531545948501733095, −12.23501717116394787353870008151, −11.29256998479924347395116550469, −10.34136347387376568552354461545, −9.481837604309626733596455589589, −8.218996315828318339050023576479, −7.28581571207042591722193271199, −5.89504023404248624054165068255, −3.92054333310681162285331929636, −1.54883955546893930442508117935, 0.51830204982355500323135113658, 3.96887949691555717663451828252, 5.88578889542697586003733196185, 6.87766132440786142398790874256, 7.69182831286356213047750860611, 9.174230401543215652974837527420, 9.818843910887307493594717207438, 11.16935461166350105603278022880, 11.55181276268556132279470672388, 13.11497178419142975577871874452

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