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Properties

Label 2-129-129.113-c1-0-5
Degree $2$
Conductor $129$
Sign $0.549 - 0.835i$
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  + (0.413 + 1.81i)2-s + (0.0986 − 1.72i)3-s + (−1.30 + 0.630i)4-s + (1.73 + 2.17i)5-s + (3.17 − 0.536i)6-s − 3.00i·7-s + (0.634 + 0.795i)8-s + (−2.98 − 0.341i)9-s + (−3.21 + 4.03i)10-s + (−1.05 + 2.18i)11-s + (0.960 + 2.32i)12-s + (1.23 + 1.55i)13-s + (5.44 − 1.24i)14-s + (3.92 − 2.77i)15-s + (−2.98 + 3.74i)16-s + (−5.95 − 4.75i)17-s + ⋯
L(s)  = 1  + (0.292 + 1.28i)2-s + (0.0569 − 0.998i)3-s + (−0.654 + 0.315i)4-s + (0.774 + 0.970i)5-s + (1.29 − 0.218i)6-s − 1.13i·7-s + (0.224 + 0.281i)8-s + (−0.993 − 0.113i)9-s + (−1.01 + 1.27i)10-s + (−0.317 + 0.659i)11-s + (0.277 + 0.671i)12-s + (0.343 + 0.430i)13-s + (1.45 − 0.331i)14-s + (1.01 − 0.717i)15-s + (−0.747 + 0.937i)16-s + (−1.44 − 1.15i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 129 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.549 - 0.835i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 129 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.549 - 0.835i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(129\)    =    \(3 \cdot 43\)
Sign: $0.549 - 0.835i$
Analytic conductor: \(1.03007\)
Root analytic conductor: \(1.01492\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{129} (113, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 129,\ (\ :1/2),\ 0.549 - 0.835i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.19926 + 0.646729i\)
\(L(\frac12)\) \(\approx\) \(1.19926 + 0.646729i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.0986 + 1.72i)T \)
43 \( 1 + (-6.31 - 1.76i)T \)
good2 \( 1 + (-0.413 - 1.81i)T + (-1.80 + 0.867i)T^{2} \)
5 \( 1 + (-1.73 - 2.17i)T + (-1.11 + 4.87i)T^{2} \)
7 \( 1 + 3.00iT - 7T^{2} \)
11 \( 1 + (1.05 - 2.18i)T + (-6.85 - 8.60i)T^{2} \)
13 \( 1 + (-1.23 - 1.55i)T + (-2.89 + 12.6i)T^{2} \)
17 \( 1 + (5.95 + 4.75i)T + (3.78 + 16.5i)T^{2} \)
19 \( 1 + (0.887 + 1.84i)T + (-11.8 + 14.8i)T^{2} \)
23 \( 1 + (-2.12 + 4.40i)T + (-14.3 - 17.9i)T^{2} \)
29 \( 1 + (-0.870 - 3.81i)T + (-26.1 + 12.5i)T^{2} \)
31 \( 1 + (1.66 + 7.30i)T + (-27.9 + 13.4i)T^{2} \)
37 \( 1 - 3.89iT - 37T^{2} \)
41 \( 1 + (9.82 - 2.24i)T + (36.9 - 17.7i)T^{2} \)
47 \( 1 + (-2.01 - 4.19i)T + (-29.3 + 36.7i)T^{2} \)
53 \( 1 + (-1.46 - 1.16i)T + (11.7 + 51.6i)T^{2} \)
59 \( 1 + (-0.961 - 0.766i)T + (13.1 + 57.5i)T^{2} \)
61 \( 1 + (-3.28 - 0.750i)T + (54.9 + 26.4i)T^{2} \)
67 \( 1 + (11.3 - 5.48i)T + (41.7 - 52.3i)T^{2} \)
71 \( 1 + (2.12 - 1.02i)T + (44.2 - 55.5i)T^{2} \)
73 \( 1 + (-3.92 + 3.13i)T + (16.2 - 71.1i)T^{2} \)
79 \( 1 - 9.47T + 79T^{2} \)
83 \( 1 + (-1.23 - 0.281i)T + (74.7 + 36.0i)T^{2} \)
89 \( 1 + (-1.73 + 7.59i)T + (-80.1 - 38.6i)T^{2} \)
97 \( 1 + (-17.1 - 8.24i)T + (60.4 + 75.8i)T^{2} \)
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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

−13.63780604220379315192047970451, −13.16960246995942982263271984543, −11.36960396730193329318929654968, −10.51336053888078710640401729036, −8.933956560495866444906738907433, −7.50053746020599431965933718927, −6.87627884675553054644584170000, −6.30395988652332924364220238737, −4.70455712102378963981586608840, −2.37218029463563054140051249652, 2.10338748210753629921188229546, 3.57323918854493164376522576138, 4.98834710354398074678627357052, 5.91079954669654829502110683406, 8.599882065991090445341082875197, 9.100663581623826600839776344653, 10.27877476765610235328020598293, 11.04225037510971692786659732027, 12.10233046315959555671984401968, 13.02248537035702788692377814372

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