Properties

Label 2-129-129.35-c0-0-0
Degree $2$
Conductor $129$
Sign $0.597 + 0.802i$
Analytic cond. $0.0643793$
Root an. cond. $0.253730$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.222 − 0.974i)3-s + (−0.900 − 0.433i)4-s + 1.24·7-s + (−0.900 + 0.433i)9-s + (−0.222 + 0.974i)12-s + (−1.12 + 1.40i)13-s + (0.623 + 0.781i)16-s + (0.400 + 0.193i)19-s + (−0.277 − 1.21i)21-s + (−0.222 − 0.974i)25-s + (0.623 + 0.781i)27-s + (−1.12 − 0.541i)28-s + (0.0990 − 0.433i)31-s + 36-s − 1.80·37-s + ⋯
L(s)  = 1  + (−0.222 − 0.974i)3-s + (−0.900 − 0.433i)4-s + 1.24·7-s + (−0.900 + 0.433i)9-s + (−0.222 + 0.974i)12-s + (−1.12 + 1.40i)13-s + (0.623 + 0.781i)16-s + (0.400 + 0.193i)19-s + (−0.277 − 1.21i)21-s + (−0.222 − 0.974i)25-s + (0.623 + 0.781i)27-s + (−1.12 − 0.541i)28-s + (0.0990 − 0.433i)31-s + 36-s − 1.80·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 129 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.597 + 0.802i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 129 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.597 + 0.802i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(129\)    =    \(3 \cdot 43\)
Sign: $0.597 + 0.802i$
Analytic conductor: \(0.0643793\)
Root analytic conductor: \(0.253730\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{129} (35, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 129,\ (\ :0),\ 0.597 + 0.802i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5239246519\)
\(L(\frac12)\) \(\approx\) \(0.5239246519\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.222 + 0.974i)T \)
43 \( 1 + (0.222 + 0.974i)T \)
good2 \( 1 + (0.900 + 0.433i)T^{2} \)
5 \( 1 + (0.222 + 0.974i)T^{2} \)
7 \( 1 - 1.24T + T^{2} \)
11 \( 1 + (-0.623 + 0.781i)T^{2} \)
13 \( 1 + (1.12 - 1.40i)T + (-0.222 - 0.974i)T^{2} \)
17 \( 1 + (0.222 - 0.974i)T^{2} \)
19 \( 1 + (-0.400 - 0.193i)T + (0.623 + 0.781i)T^{2} \)
23 \( 1 + (-0.623 + 0.781i)T^{2} \)
29 \( 1 + (0.900 + 0.433i)T^{2} \)
31 \( 1 + (-0.0990 + 0.433i)T + (-0.900 - 0.433i)T^{2} \)
37 \( 1 + 1.80T + T^{2} \)
41 \( 1 + (0.900 + 0.433i)T^{2} \)
47 \( 1 + (-0.623 - 0.781i)T^{2} \)
53 \( 1 + (0.222 - 0.974i)T^{2} \)
59 \( 1 + (0.222 - 0.974i)T^{2} \)
61 \( 1 + (-0.400 - 1.75i)T + (-0.900 + 0.433i)T^{2} \)
67 \( 1 + (1.12 + 0.541i)T + (0.623 + 0.781i)T^{2} \)
71 \( 1 + (-0.623 - 0.781i)T^{2} \)
73 \( 1 + (0.277 - 0.347i)T + (-0.222 - 0.974i)T^{2} \)
79 \( 1 - 1.24T + T^{2} \)
83 \( 1 + (0.900 - 0.433i)T^{2} \)
89 \( 1 + (0.900 - 0.433i)T^{2} \)
97 \( 1 + (1.12 - 0.541i)T + (0.623 - 0.781i)T^{2} \)
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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

−13.71229990085428705274954435277, −12.28964515735597907130185865585, −11.65469192889344300314526221735, −10.37997946206077178421736917455, −9.052437743096030447500602284468, −8.100670229745566778273229219911, −6.96923331228537653412714555091, −5.49395294756135925311833737007, −4.49013472396186455404207866166, −1.86080065856534777136795850297, 3.30732714755598213119566821887, 4.80995547328830643512964708725, 5.33303418764230510184900226922, 7.61301906145116216345331273481, 8.514268372219487643025238323980, 9.601672425406182691991039024219, 10.54474617737840929322936357335, 11.65046371960817269559606772409, 12.62467929145283252100441162066, 13.91927817308950055824431680332

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