Properties

Label 2-129-129.2-c1-0-3
Degree $2$
Conductor $129$
Sign $-0.333 - 0.942i$
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  + (−1.79 + 0.866i)2-s + (1.60 + 0.653i)3-s + (1.23 − 1.55i)4-s + (−0.490 + 2.15i)5-s + (−3.45 + 0.214i)6-s + 0.451i·7-s + (0.00561 − 0.0245i)8-s + (2.14 + 2.09i)9-s + (−0.979 − 4.29i)10-s + (−0.584 + 0.466i)11-s + (3.00 − 1.68i)12-s + (−0.438 + 1.92i)13-s + (−0.391 − 0.813i)14-s + (−2.19 + 3.12i)15-s + (0.895 + 3.92i)16-s + (−5.03 + 1.14i)17-s + ⋯
L(s)  = 1  + (−1.27 + 0.612i)2-s + (0.926 + 0.377i)3-s + (0.619 − 0.776i)4-s + (−0.219 + 0.961i)5-s + (−1.40 + 0.0875i)6-s + 0.170i·7-s + (0.00198 − 0.00869i)8-s + (0.715 + 0.698i)9-s + (−0.309 − 1.35i)10-s + (−0.176 + 0.140i)11-s + (0.866 − 0.485i)12-s + (−0.121 + 0.532i)13-s + (−0.104 − 0.217i)14-s + (−0.565 + 0.807i)15-s + (0.223 + 0.981i)16-s + (−1.22 + 0.278i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 129 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.333 - 0.942i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 129 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.333 - 0.942i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(129\)    =    \(3 \cdot 43\)
Sign: $-0.333 - 0.942i$
Analytic conductor: \(1.03007\)
Root analytic conductor: \(1.01492\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{129} (2, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 129,\ (\ :1/2),\ -0.333 - 0.942i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.420330 + 0.594531i\)
\(L(\frac12)\) \(\approx\) \(0.420330 + 0.594531i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.60 - 0.653i)T \)
43 \( 1 + (2.29 + 6.14i)T \)
good2 \( 1 + (1.79 - 0.866i)T + (1.24 - 1.56i)T^{2} \)
5 \( 1 + (0.490 - 2.15i)T + (-4.50 - 2.16i)T^{2} \)
7 \( 1 - 0.451iT - 7T^{2} \)
11 \( 1 + (0.584 - 0.466i)T + (2.44 - 10.7i)T^{2} \)
13 \( 1 + (0.438 - 1.92i)T + (-11.7 - 5.64i)T^{2} \)
17 \( 1 + (5.03 - 1.14i)T + (15.3 - 7.37i)T^{2} \)
19 \( 1 + (1.09 + 0.875i)T + (4.22 + 18.5i)T^{2} \)
23 \( 1 + (-6.31 + 5.03i)T + (5.11 - 22.4i)T^{2} \)
29 \( 1 + (-8.19 + 3.94i)T + (18.0 - 22.6i)T^{2} \)
31 \( 1 + (-0.500 + 0.241i)T + (19.3 - 24.2i)T^{2} \)
37 \( 1 + 9.70iT - 37T^{2} \)
41 \( 1 + (-0.854 - 1.77i)T + (-25.5 + 32.0i)T^{2} \)
47 \( 1 + (-4.00 - 3.19i)T + (10.4 + 45.8i)T^{2} \)
53 \( 1 + (6.92 - 1.58i)T + (47.7 - 22.9i)T^{2} \)
59 \( 1 + (-5.16 + 1.17i)T + (53.1 - 25.5i)T^{2} \)
61 \( 1 + (1.02 - 2.11i)T + (-38.0 - 47.6i)T^{2} \)
67 \( 1 + (5.86 - 7.35i)T + (-14.9 - 65.3i)T^{2} \)
71 \( 1 + (-6.65 + 8.34i)T + (-15.7 - 69.2i)T^{2} \)
73 \( 1 + (13.4 + 3.06i)T + (65.7 + 31.6i)T^{2} \)
79 \( 1 - 10.8T + 79T^{2} \)
83 \( 1 + (4.68 - 9.71i)T + (-51.7 - 64.8i)T^{2} \)
89 \( 1 + (11.2 + 5.42i)T + (55.4 + 69.5i)T^{2} \)
97 \( 1 + (4.86 + 6.10i)T + (-21.5 + 94.5i)T^{2} \)
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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

−13.96038395110580706152893202516, −12.76200866482373564352806236811, −10.96459529873481646705917698339, −10.33958202230307493687571003463, −9.144639112045501674796723411017, −8.530006719603129334248073899779, −7.32087010958783041537620381110, −6.61230806794409175071254376921, −4.30140105087043457938737382557, −2.55011080516328571288693070383, 1.17390064964466610583210992449, 2.88045678065087130578497279492, 4.80828154495992728161856861631, 7.00586281362474598903977594289, 8.212829015742549517325539031813, 8.739966755605789584696520037526, 9.623527240265015270276913312949, 10.72373969310850373190623725446, 11.92621736327223820936907181264, 12.92250049572604459104222175338

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