Properties

Label 2-129-43.4-c1-0-1
Degree $2$
Conductor $129$
Sign $0.210 - 0.977i$
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  + (−0.777 + 0.974i)2-s + (−0.623 − 0.781i)3-s + (0.0990 + 0.433i)4-s + (1.72 + 0.829i)5-s + 1.24·6-s + 1.35·7-s + (−2.74 − 1.32i)8-s + (−0.222 + 0.974i)9-s + (−2.14 + 1.03i)10-s + (−1.30 + 5.70i)11-s + (0.277 − 0.347i)12-s + (5.35 + 2.57i)13-s + (−1.05 + 1.32i)14-s + (−0.425 − 1.86i)15-s + (2.62 − 1.26i)16-s + (−2.24 + 1.08i)17-s + ⋯
L(s)  = 1  + (−0.549 + 0.689i)2-s + (−0.359 − 0.451i)3-s + (0.0495 + 0.216i)4-s + (0.770 + 0.370i)5-s + 0.509·6-s + 0.512·7-s + (−0.971 − 0.467i)8-s + (−0.0741 + 0.324i)9-s + (−0.679 + 0.327i)10-s + (−0.392 + 1.71i)11-s + (0.0801 − 0.100i)12-s + (1.48 + 0.714i)13-s + (−0.281 + 0.353i)14-s + (−0.109 − 0.481i)15-s + (0.655 − 0.315i)16-s + (−0.544 + 0.262i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 129 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.210 - 0.977i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 129 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.210 - 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(129\)    =    \(3 \cdot 43\)
Sign: $0.210 - 0.977i$
Analytic conductor: \(1.03007\)
Root analytic conductor: \(1.01492\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{129} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 129,\ (\ :1/2),\ 0.210 - 0.977i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.661440 + 0.534139i\)
\(L(\frac12)\) \(\approx\) \(0.661440 + 0.534139i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.623 + 0.781i)T \)
43 \( 1 + (6.24 - 1.99i)T \)
good2 \( 1 + (0.777 - 0.974i)T + (-0.445 - 1.94i)T^{2} \)
5 \( 1 + (-1.72 - 0.829i)T + (3.11 + 3.90i)T^{2} \)
7 \( 1 - 1.35T + 7T^{2} \)
11 \( 1 + (1.30 - 5.70i)T + (-9.91 - 4.77i)T^{2} \)
13 \( 1 + (-5.35 - 2.57i)T + (8.10 + 10.1i)T^{2} \)
17 \( 1 + (2.24 - 1.08i)T + (10.5 - 13.2i)T^{2} \)
19 \( 1 + (1.23 + 5.41i)T + (-17.1 + 8.24i)T^{2} \)
23 \( 1 + (-1.88 + 8.24i)T + (-20.7 - 9.97i)T^{2} \)
29 \( 1 + (-2.10 + 2.64i)T + (-6.45 - 28.2i)T^{2} \)
31 \( 1 + (1.77 - 2.22i)T + (-6.89 - 30.2i)T^{2} \)
37 \( 1 - 4.78T + 37T^{2} \)
41 \( 1 + (0.538 - 0.674i)T + (-9.12 - 39.9i)T^{2} \)
47 \( 1 + (0.228 + 1.00i)T + (-42.3 + 20.3i)T^{2} \)
53 \( 1 + (-10.2 + 4.92i)T + (33.0 - 41.4i)T^{2} \)
59 \( 1 + (3.65 - 1.76i)T + (36.7 - 46.1i)T^{2} \)
61 \( 1 + (2.71 + 3.40i)T + (-13.5 + 59.4i)T^{2} \)
67 \( 1 + (2.43 + 10.6i)T + (-60.3 + 29.0i)T^{2} \)
71 \( 1 + (0.0658 + 0.288i)T + (-63.9 + 30.8i)T^{2} \)
73 \( 1 + (-6.35 - 3.06i)T + (45.5 + 57.0i)T^{2} \)
79 \( 1 + 4.55T + 79T^{2} \)
83 \( 1 + (3.43 + 4.30i)T + (-18.4 + 80.9i)T^{2} \)
89 \( 1 + (-2.45 - 3.07i)T + (-19.8 + 86.7i)T^{2} \)
97 \( 1 + (3.41 - 14.9i)T + (-87.3 - 42.0i)T^{2} \)
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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

−13.44317631393696563056686010620, −12.65146254314257118109603982929, −11.47714854934710650360732874748, −10.38692609013698572332218989278, −9.124141193860589424364003121330, −8.149406182198284233513954425860, −6.84168964788404753882044442594, −6.38120679614952777854097342877, −4.60287756590835366947682183898, −2.24236084514485675507767684105, 1.32004901995573857428150453640, 3.37195381252403208855384968488, 5.55771703745341913511390894906, 5.91827895703635024856037560151, 8.275604218764341849474530600048, 9.053927277322688287352832217653, 10.17346217835495973895425552603, 11.01616210325992882919996353933, 11.53846729260139921450778786506, 13.15735966480111001676827743743

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