Properties

Label 2-138-23.8-c1-0-1
Degree $2$
Conductor $138$
Sign $0.677 - 0.735i$
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  + (−0.142 + 0.989i)2-s + (−0.415 − 0.909i)3-s + (−0.959 − 0.281i)4-s + (0.698 + 0.806i)5-s + (0.959 − 0.281i)6-s + (3.72 + 2.39i)7-s + (0.415 − 0.909i)8-s + (−0.654 + 0.755i)9-s + (−0.897 + 0.577i)10-s + (0.234 + 1.63i)11-s + (0.142 + 0.989i)12-s + (4.60 − 2.95i)13-s + (−2.89 + 3.34i)14-s + (0.443 − 0.970i)15-s + (0.841 + 0.540i)16-s + (−3.76 + 1.10i)17-s + ⋯
L(s)  = 1  + (−0.100 + 0.699i)2-s + (−0.239 − 0.525i)3-s + (−0.479 − 0.140i)4-s + (0.312 + 0.360i)5-s + (0.391 − 0.115i)6-s + (1.40 + 0.904i)7-s + (0.146 − 0.321i)8-s + (−0.218 + 0.251i)9-s + (−0.283 + 0.182i)10-s + (0.0707 + 0.492i)11-s + (0.0410 + 0.285i)12-s + (1.27 − 0.820i)13-s + (−0.774 + 0.893i)14-s + (0.114 − 0.250i)15-s + (0.210 + 0.135i)16-s + (−0.913 + 0.268i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.677 - 0.735i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.677 - 0.735i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(138\)    =    \(2 \cdot 3 \cdot 23\)
Sign: $0.677 - 0.735i$
Analytic conductor: \(1.10193\)
Root analytic conductor: \(1.04973\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{138} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 138,\ (\ :1/2),\ 0.677 - 0.735i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.982358 + 0.430531i\)
\(L(\frac12)\) \(\approx\) \(0.982358 + 0.430531i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.142 - 0.989i)T \)
3 \( 1 + (0.415 + 0.909i)T \)
23 \( 1 + (4.78 + 0.323i)T \)
good5 \( 1 + (-0.698 - 0.806i)T + (-0.711 + 4.94i)T^{2} \)
7 \( 1 + (-3.72 - 2.39i)T + (2.90 + 6.36i)T^{2} \)
11 \( 1 + (-0.234 - 1.63i)T + (-10.5 + 3.09i)T^{2} \)
13 \( 1 + (-4.60 + 2.95i)T + (5.40 - 11.8i)T^{2} \)
17 \( 1 + (3.76 - 1.10i)T + (14.3 - 9.19i)T^{2} \)
19 \( 1 + (6.33 + 1.85i)T + (15.9 + 10.2i)T^{2} \)
29 \( 1 + (-6.14 + 1.80i)T + (24.3 - 15.6i)T^{2} \)
31 \( 1 + (1.25 - 2.74i)T + (-20.3 - 23.4i)T^{2} \)
37 \( 1 + (-1.44 + 1.67i)T + (-5.26 - 36.6i)T^{2} \)
41 \( 1 + (2.67 + 3.09i)T + (-5.83 + 40.5i)T^{2} \)
43 \( 1 + (1.21 + 2.64i)T + (-28.1 + 32.4i)T^{2} \)
47 \( 1 + 9.62T + 47T^{2} \)
53 \( 1 + (1.87 + 1.20i)T + (22.0 + 48.2i)T^{2} \)
59 \( 1 + (-8.20 + 5.27i)T + (24.5 - 53.6i)T^{2} \)
61 \( 1 + (0.647 - 1.41i)T + (-39.9 - 46.1i)T^{2} \)
67 \( 1 + (1.71 - 11.9i)T + (-64.2 - 18.8i)T^{2} \)
71 \( 1 + (-0.749 + 5.21i)T + (-68.1 - 20.0i)T^{2} \)
73 \( 1 + (6.89 + 2.02i)T + (61.4 + 39.4i)T^{2} \)
79 \( 1 + (-2.26 + 1.45i)T + (32.8 - 71.8i)T^{2} \)
83 \( 1 + (3.62 - 4.18i)T + (-11.8 - 82.1i)T^{2} \)
89 \( 1 + (-4.70 - 10.3i)T + (-58.2 + 67.2i)T^{2} \)
97 \( 1 + (7.84 + 9.05i)T + (-13.8 + 96.0i)T^{2} \)
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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

−13.43328773236877304004880631744, −12.41729474109219932409144843832, −11.26399645911647329017060567023, −10.36440987977612283933737326605, −8.557428479630086110325286869648, −8.264233353990898729008714900055, −6.68146448524100227242404020360, −5.83411941093625982145067564158, −4.55646159590577678250025061701, −2.06107111598418183696851393998, 1.60348294142352999779041223287, 3.95403265836380024578165053345, 4.76973497030798439490305308654, 6.34746652084959767542173891480, 8.175595574823469265691906417456, 8.893878714161657077552350837240, 10.25220130920474447014060194275, 11.09866957460464632849491595057, 11.59861340891831015000962308433, 13.14924724539727027220956362458

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