Properties

Label 2-138-23.3-c1-0-0
Degree $2$
Conductor $138$
Sign $-0.741 - 0.670i$
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 + (−1.81 + 2.09i)5-s + (−0.959 − 0.281i)6-s + (0.163 − 0.105i)7-s + (−0.415 − 0.909i)8-s + (−0.654 − 0.755i)9-s + (−2.33 − 1.49i)10-s + (−0.413 + 2.87i)11-s + (0.142 − 0.989i)12-s + (0.910 + 0.585i)13-s + (0.127 + 0.146i)14-s + (−1.15 − 2.52i)15-s + (0.841 − 0.540i)16-s + (4.45 + 1.30i)17-s + ⋯
L(s)  = 1  + (0.100 + 0.699i)2-s + (−0.239 + 0.525i)3-s + (−0.479 + 0.140i)4-s + (−0.811 + 0.936i)5-s + (−0.391 − 0.115i)6-s + (0.0617 − 0.0396i)7-s + (−0.146 − 0.321i)8-s + (−0.218 − 0.251i)9-s + (−0.737 − 0.473i)10-s + (−0.124 + 0.866i)11-s + (0.0410 − 0.285i)12-s + (0.252 + 0.162i)13-s + (0.0339 + 0.0392i)14-s + (−0.297 − 0.650i)15-s + (0.210 − 0.135i)16-s + (1.08 + 0.317i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.741 - 0.670i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.741 - 0.670i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(138\)    =    \(2 \cdot 3 \cdot 23\)
Sign: $-0.741 - 0.670i$
Analytic conductor: \(1.10193\)
Root analytic conductor: \(1.04973\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{138} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 138,\ (\ :1/2),\ -0.741 - 0.670i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.310919 + 0.807764i\)
\(L(\frac12)\) \(\approx\) \(0.310919 + 0.807764i\)
\(L(\frac{3}{2})\) 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 \( 1 + (4.73 - 0.757i)T \)
good5 \( 1 + (1.81 - 2.09i)T + (-0.711 - 4.94i)T^{2} \)
7 \( 1 + (-0.163 + 0.105i)T + (2.90 - 6.36i)T^{2} \)
11 \( 1 + (0.413 - 2.87i)T + (-10.5 - 3.09i)T^{2} \)
13 \( 1 + (-0.910 - 0.585i)T + (5.40 + 11.8i)T^{2} \)
17 \( 1 + (-4.45 - 1.30i)T + (14.3 + 9.19i)T^{2} \)
19 \( 1 + (-4.24 + 1.24i)T + (15.9 - 10.2i)T^{2} \)
29 \( 1 + (-8.83 - 2.59i)T + (24.3 + 15.6i)T^{2} \)
31 \( 1 + (2.85 + 6.25i)T + (-20.3 + 23.4i)T^{2} \)
37 \( 1 + (-3.17 - 3.66i)T + (-5.26 + 36.6i)T^{2} \)
41 \( 1 + (-3.76 + 4.34i)T + (-5.83 - 40.5i)T^{2} \)
43 \( 1 + (-3.14 + 6.88i)T + (-28.1 - 32.4i)T^{2} \)
47 \( 1 + 5.61T + 47T^{2} \)
53 \( 1 + (3.48 - 2.23i)T + (22.0 - 48.2i)T^{2} \)
59 \( 1 + (9.01 + 5.79i)T + (24.5 + 53.6i)T^{2} \)
61 \( 1 + (1.19 + 2.61i)T + (-39.9 + 46.1i)T^{2} \)
67 \( 1 + (-2.08 - 14.4i)T + (-64.2 + 18.8i)T^{2} \)
71 \( 1 + (1.81 + 12.6i)T + (-68.1 + 20.0i)T^{2} \)
73 \( 1 + (2.86 - 0.841i)T + (61.4 - 39.4i)T^{2} \)
79 \( 1 + (-2.15 - 1.38i)T + (32.8 + 71.8i)T^{2} \)
83 \( 1 + (-0.531 - 0.613i)T + (-11.8 + 82.1i)T^{2} \)
89 \( 1 + (-2.81 + 6.16i)T + (-58.2 - 67.2i)T^{2} \)
97 \( 1 + (-8.09 + 9.34i)T + (-13.8 - 96.0i)T^{2} \)
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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

−13.97646676471361403578080768752, −12.44854416126228092282495356959, −11.57147006560804487326066538626, −10.44103685618951582477742888137, −9.519218077804436595782125333060, −7.996802543237717244077371433565, −7.21219480572979097330422854589, −5.96494471133739929134685424002, −4.55348943232094850890617485398, −3.33733789944420783868441498755, 0.980856827806809260870526227219, 3.25861174126809108792451682236, 4.75495145381039917345914694195, 5.95555070429888273802535979893, 7.79592252407485357542558939765, 8.453932190353415760640027169907, 9.786693720069600306852538048255, 11.06818653510846912104025811763, 11.99487199290513926926868916920, 12.46572957227381424076420600210

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