Properties

Label 2-138-23.9-c1-0-2
Degree $2$
Conductor $138$
Sign $0.999 + 0.0268i$
Analytic cond. $1.10193$
Root an. cond. $1.04973$
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  + (0.959 − 0.281i)2-s + (0.654 + 0.755i)3-s + (0.841 − 0.540i)4-s + (−0.0651 − 0.453i)5-s + (0.841 + 0.540i)6-s + (−0.134 + 0.295i)7-s + (0.654 − 0.755i)8-s + (−0.142 + 0.989i)9-s + (−0.190 − 0.416i)10-s + (−2.38 − 0.701i)11-s + (0.959 + 0.281i)12-s + (0.564 + 1.23i)13-s + (−0.0462 + 0.321i)14-s + (0.299 − 0.346i)15-s + (0.415 − 0.909i)16-s + (−2.26 − 1.45i)17-s + ⋯
L(s)  = 1  + (0.678 − 0.199i)2-s + (0.378 + 0.436i)3-s + (0.420 − 0.270i)4-s + (−0.0291 − 0.202i)5-s + (0.343 + 0.220i)6-s + (−0.0509 + 0.111i)7-s + (0.231 − 0.267i)8-s + (−0.0474 + 0.329i)9-s + (−0.0601 − 0.131i)10-s + (−0.719 − 0.211i)11-s + (0.276 + 0.0813i)12-s + (0.156 + 0.342i)13-s + (−0.0123 + 0.0858i)14-s + (0.0774 − 0.0893i)15-s + (0.103 − 0.227i)16-s + (−0.550 − 0.353i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0268i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0268i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(138\)    =    \(2 \cdot 3 \cdot 23\)
Sign: $0.999 + 0.0268i$
Analytic conductor: \(1.10193\)
Root analytic conductor: \(1.04973\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{138} (55, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 138,\ (\ :1/2),\ 0.999 + 0.0268i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.64765 - 0.0221112i\)
\(L(\frac12)\) \(\approx\) \(1.64765 - 0.0221112i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.959 + 0.281i)T \)
3 \( 1 + (-0.654 - 0.755i)T \)
23 \( 1 + (4.35 + 2.00i)T \)
good5 \( 1 + (0.0651 + 0.453i)T + (-4.79 + 1.40i)T^{2} \)
7 \( 1 + (0.134 - 0.295i)T + (-4.58 - 5.29i)T^{2} \)
11 \( 1 + (2.38 + 0.701i)T + (9.25 + 5.94i)T^{2} \)
13 \( 1 + (-0.564 - 1.23i)T + (-8.51 + 9.82i)T^{2} \)
17 \( 1 + (2.26 + 1.45i)T + (7.06 + 15.4i)T^{2} \)
19 \( 1 + (2.43 - 1.56i)T + (7.89 - 17.2i)T^{2} \)
29 \( 1 + (1.83 + 1.17i)T + (12.0 + 26.3i)T^{2} \)
31 \( 1 + (1.61 - 1.86i)T + (-4.41 - 30.6i)T^{2} \)
37 \( 1 + (0.0683 - 0.475i)T + (-35.5 - 10.4i)T^{2} \)
41 \( 1 + (0.120 + 0.838i)T + (-39.3 + 11.5i)T^{2} \)
43 \( 1 + (-5.76 - 6.65i)T + (-6.11 + 42.5i)T^{2} \)
47 \( 1 + 2.71T + 47T^{2} \)
53 \( 1 + (-3.62 + 7.93i)T + (-34.7 - 40.0i)T^{2} \)
59 \( 1 + (-4.52 - 9.89i)T + (-38.6 + 44.5i)T^{2} \)
61 \( 1 + (-7.89 + 9.11i)T + (-8.68 - 60.3i)T^{2} \)
67 \( 1 + (10.1 - 2.97i)T + (56.3 - 36.2i)T^{2} \)
71 \( 1 + (-15.7 + 4.62i)T + (59.7 - 38.3i)T^{2} \)
73 \( 1 + (-9.89 + 6.35i)T + (30.3 - 66.4i)T^{2} \)
79 \( 1 + (4.45 + 9.75i)T + (-51.7 + 59.7i)T^{2} \)
83 \( 1 + (1.15 - 8.00i)T + (-79.6 - 23.3i)T^{2} \)
89 \( 1 + (-6.74 - 7.78i)T + (-12.6 + 88.0i)T^{2} \)
97 \( 1 + (-0.0319 - 0.221i)T + (-93.0 + 27.3i)T^{2} \)
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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

−13.22271022879247047530606205453, −12.33682275752013643127784174976, −11.14104949179606379915222884452, −10.28209479343438458359229527919, −9.064594894979581372436338074545, −7.946805987342798098087675154956, −6.45677070662978907198268362054, −5.12893402197946579554911042519, −3.98125946578546249138041784781, −2.46115382359368074700392390996, 2.39725447086874115817248376757, 3.90043981022277574650530208845, 5.43773865802009346703657157176, 6.70486749085688156525665364724, 7.68767459575263050383799525693, 8.783340976292928377836286459246, 10.29297011654148769958862821478, 11.28743602867960557798794017649, 12.55538218199183323521418130206, 13.16869648491521545838283789784

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