Properties

Label 2-129-129.89-c1-0-10
Degree $2$
Conductor $129$
Sign $0.860 - 0.509i$
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  + (1.83 + 0.882i)2-s + (1.70 − 0.313i)3-s + (1.33 + 1.67i)4-s + (−2.87 − 0.886i)5-s + (3.39 + 0.928i)6-s + (−3.63 + 2.09i)7-s + (0.0631 + 0.276i)8-s + (2.80 − 1.06i)9-s + (−4.48 − 4.16i)10-s + (−0.919 − 0.733i)11-s + (2.79 + 2.43i)12-s + (2.60 − 2.41i)13-s + (−8.50 + 0.637i)14-s + (−5.17 − 0.608i)15-s + (0.823 − 3.60i)16-s + (1.98 + 6.42i)17-s + ⋯
L(s)  = 1  + (1.29 + 0.624i)2-s + (0.983 − 0.181i)3-s + (0.666 + 0.836i)4-s + (−1.28 − 0.396i)5-s + (1.38 + 0.379i)6-s + (−1.37 + 0.792i)7-s + (0.0223 + 0.0977i)8-s + (0.934 − 0.356i)9-s + (−1.41 − 1.31i)10-s + (−0.277 − 0.221i)11-s + (0.807 + 0.701i)12-s + (0.722 − 0.670i)13-s + (−2.27 + 0.170i)14-s + (−1.33 − 0.157i)15-s + (0.205 − 0.902i)16-s + (0.480 + 1.55i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 129 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.860 - 0.509i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 129 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.860 - 0.509i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(129\)    =    \(3 \cdot 43\)
Sign: $0.860 - 0.509i$
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: \((2,\ 129,\ (\ :1/2),\ 0.860 - 0.509i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.94116 + 0.531599i\)
\(L(\frac12)\) \(\approx\) \(1.94116 + 0.531599i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.70 + 0.313i)T \)
43 \( 1 + (2.96 + 5.84i)T \)
good2 \( 1 + (-1.83 - 0.882i)T + (1.24 + 1.56i)T^{2} \)
5 \( 1 + (2.87 + 0.886i)T + (4.13 + 2.81i)T^{2} \)
7 \( 1 + (3.63 - 2.09i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (0.919 + 0.733i)T + (2.44 + 10.7i)T^{2} \)
13 \( 1 + (-2.60 + 2.41i)T + (0.971 - 12.9i)T^{2} \)
17 \( 1 + (-1.98 - 6.42i)T + (-14.0 + 9.57i)T^{2} \)
19 \( 1 + (-0.400 - 2.65i)T + (-18.1 + 5.60i)T^{2} \)
23 \( 1 + (-0.995 + 0.390i)T + (16.8 - 15.6i)T^{2} \)
29 \( 1 + (-0.272 - 3.63i)T + (-28.6 + 4.32i)T^{2} \)
31 \( 1 + (1.18 - 0.810i)T + (11.3 - 28.8i)T^{2} \)
37 \( 1 + (6.95 + 4.01i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (0.391 - 0.812i)T + (-25.5 - 32.0i)T^{2} \)
47 \( 1 + (3.09 - 2.47i)T + (10.4 - 45.8i)T^{2} \)
53 \( 1 + (-9.47 + 10.2i)T + (-3.96 - 52.8i)T^{2} \)
59 \( 1 + (8.23 + 1.87i)T + (53.1 + 25.5i)T^{2} \)
61 \( 1 + (2.60 - 3.81i)T + (-22.2 - 56.7i)T^{2} \)
67 \( 1 + (1.02 - 0.154i)T + (64.0 - 19.7i)T^{2} \)
71 \( 1 + (2.72 - 6.93i)T + (-52.0 - 48.2i)T^{2} \)
73 \( 1 + (-7.57 - 8.16i)T + (-5.45 + 72.7i)T^{2} \)
79 \( 1 + (1.16 + 2.02i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-6.55 - 0.491i)T + (82.0 + 12.3i)T^{2} \)
89 \( 1 + (0.318 - 4.24i)T + (-88.0 - 13.2i)T^{2} \)
97 \( 1 + (-7.18 + 9.00i)T + (-21.5 - 94.5i)T^{2} \)
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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

−13.27718842006721797339944296443, −12.67036450630841293910995234816, −12.16471383453656499834437823516, −10.26470037191785501293855476801, −8.812682646233858532390129190181, −7.966045543765631194585242057875, −6.72820537485330497211155905711, −5.57929426221926605337385891648, −3.80330005447416853527103950312, −3.32857766767494514666179631391, 2.97513359144280548313721774320, 3.62609902461213289082978815718, 4.62305925937543552474639896397, 6.72256986394283447939889501101, 7.67003455037890408672852041462, 9.175524466120530162051483282493, 10.38133582592896012669359865945, 11.45728810810223688103823191855, 12.37922905899288684996933454299, 13.57108117650598945219595034163

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