Properties

Label 2-138-23.6-c1-0-0
Degree $2$
Conductor $138$
Sign $0.999 + 0.0348i$
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  + (−0.415 + 0.909i)2-s + (−0.959 − 0.281i)3-s + (−0.654 − 0.755i)4-s + (2.43 − 1.56i)5-s + (0.654 − 0.755i)6-s + (0.394 − 2.74i)7-s + (0.959 − 0.281i)8-s + (0.841 + 0.540i)9-s + (0.411 + 2.86i)10-s + (2.26 + 4.95i)11-s + (0.415 + 0.909i)12-s + (−0.0520 − 0.361i)13-s + (2.33 + 1.49i)14-s + (−2.77 + 0.815i)15-s + (−0.142 + 0.989i)16-s + (4.12 − 4.75i)17-s + ⋯
L(s)  = 1  + (−0.293 + 0.643i)2-s + (−0.553 − 0.162i)3-s + (−0.327 − 0.377i)4-s + (1.08 − 0.699i)5-s + (0.267 − 0.308i)6-s + (0.149 − 1.03i)7-s + (0.339 − 0.0996i)8-s + (0.280 + 0.180i)9-s + (0.130 + 0.905i)10-s + (0.682 + 1.49i)11-s + (0.119 + 0.262i)12-s + (−0.0144 − 0.100i)13-s + (0.622 + 0.400i)14-s + (−0.716 + 0.210i)15-s + (−0.0355 + 0.247i)16-s + (1.00 − 1.15i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0348i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0348i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(138\)    =    \(2 \cdot 3 \cdot 23\)
Sign: $0.999 + 0.0348i$
Analytic conductor: \(1.10193\)
Root analytic conductor: \(1.04973\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{138} (121, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 138,\ (\ :1/2),\ 0.999 + 0.0348i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.943931 - 0.0164483i\)
\(L(\frac12)\) \(\approx\) \(0.943931 - 0.0164483i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.415 - 0.909i)T \)
3 \( 1 + (0.959 + 0.281i)T \)
23 \( 1 + (0.965 - 4.69i)T \)
good5 \( 1 + (-2.43 + 1.56i)T + (2.07 - 4.54i)T^{2} \)
7 \( 1 + (-0.394 + 2.74i)T + (-6.71 - 1.97i)T^{2} \)
11 \( 1 + (-2.26 - 4.95i)T + (-7.20 + 8.31i)T^{2} \)
13 \( 1 + (0.0520 + 0.361i)T + (-12.4 + 3.66i)T^{2} \)
17 \( 1 + (-4.12 + 4.75i)T + (-2.41 - 16.8i)T^{2} \)
19 \( 1 + (4.01 + 4.63i)T + (-2.70 + 18.8i)T^{2} \)
29 \( 1 + (2.23 - 2.57i)T + (-4.12 - 28.7i)T^{2} \)
31 \( 1 + (7.37 - 2.16i)T + (26.0 - 16.7i)T^{2} \)
37 \( 1 + (-4.06 - 2.61i)T + (15.3 + 33.6i)T^{2} \)
41 \( 1 + (3.56 - 2.29i)T + (17.0 - 37.2i)T^{2} \)
43 \( 1 + (-6.75 - 1.98i)T + (36.1 + 23.2i)T^{2} \)
47 \( 1 + 6.68T + 47T^{2} \)
53 \( 1 + (1.01 - 7.06i)T + (-50.8 - 14.9i)T^{2} \)
59 \( 1 + (0.0626 + 0.435i)T + (-56.6 + 16.6i)T^{2} \)
61 \( 1 + (7.80 - 2.29i)T + (51.3 - 32.9i)T^{2} \)
67 \( 1 + (-1.84 + 4.04i)T + (-43.8 - 50.6i)T^{2} \)
71 \( 1 + (0.586 - 1.28i)T + (-46.4 - 53.6i)T^{2} \)
73 \( 1 + (-2.58 - 2.97i)T + (-10.3 + 72.2i)T^{2} \)
79 \( 1 + (-0.565 - 3.93i)T + (-75.7 + 22.2i)T^{2} \)
83 \( 1 + (-13.5 - 8.70i)T + (34.4 + 75.4i)T^{2} \)
89 \( 1 + (13.2 + 3.87i)T + (74.8 + 48.1i)T^{2} \)
97 \( 1 + (3.64 - 2.34i)T + (40.2 - 88.2i)T^{2} \)
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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

−13.27368763559427215478645651759, −12.38252467415475388395065921284, −10.98397770023013197635222334468, −9.778987787933814135862995444552, −9.296438450697313142122560257042, −7.55043159816356926677255863212, −6.79914770647445511989703762645, −5.43953355367735571031610232312, −4.50089422341723099114902884208, −1.44568029499968884435664446617, 2.01235839338671016318229200224, 3.65935103232670695353050439161, 5.75851768230333562042675882712, 6.19856333974403990951293464482, 8.228967587873290978809285033067, 9.235760121773541026273125859600, 10.31317361972982274125412002501, 11.00658400812590545978678489518, 12.05868275639747913331384504287, 12.94639085085242238334749969267

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