Properties

Label 2-138-23.9-c3-0-7
Degree $2$
Conductor $138$
Sign $0.595 + 0.803i$
Analytic cond. $8.14226$
Root an. cond. $2.85346$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.91 − 0.563i)2-s + (−1.96 − 2.26i)3-s + (3.36 − 2.16i)4-s + (1.75 + 12.2i)5-s + (−5.04 − 3.24i)6-s + (12.8 − 28.1i)7-s + (5.23 − 6.04i)8-s + (−1.28 + 8.90i)9-s + (10.2 + 22.4i)10-s + (38.1 + 11.2i)11-s + (−11.5 − 3.38i)12-s + (−28.5 − 62.4i)13-s + (8.81 − 61.2i)14-s + (24.2 − 27.9i)15-s + (6.64 − 14.5i)16-s + (73.7 + 47.3i)17-s + ⋯
L(s)  = 1  + (0.678 − 0.199i)2-s + (−0.378 − 0.436i)3-s + (0.420 − 0.270i)4-s + (0.157 + 1.09i)5-s + (−0.343 − 0.220i)6-s + (0.694 − 1.52i)7-s + (0.231 − 0.267i)8-s + (−0.0474 + 0.329i)9-s + (0.324 + 0.710i)10-s + (1.04 + 0.307i)11-s + (−0.276 − 0.0813i)12-s + (−0.608 − 1.33i)13-s + (0.168 − 1.16i)14-s + (0.417 − 0.481i)15-s + (0.103 − 0.227i)16-s + (1.05 + 0.676i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.595 + 0.803i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.595 + 0.803i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(138\)    =    \(2 \cdot 3 \cdot 23\)
Sign: $0.595 + 0.803i$
Analytic conductor: \(8.14226\)
Root analytic conductor: \(2.85346\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{138} (55, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 138,\ (\ :3/2),\ 0.595 + 0.803i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.16715 - 1.09157i\)
\(L(\frac12)\) \(\approx\) \(2.16715 - 1.09157i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.91 + 0.563i)T \)
3 \( 1 + (1.96 + 2.26i)T \)
23 \( 1 + (25.3 + 107. i)T \)
good5 \( 1 + (-1.75 - 12.2i)T + (-119. + 35.2i)T^{2} \)
7 \( 1 + (-12.8 + 28.1i)T + (-224. - 259. i)T^{2} \)
11 \( 1 + (-38.1 - 11.2i)T + (1.11e3 + 719. i)T^{2} \)
13 \( 1 + (28.5 + 62.4i)T + (-1.43e3 + 1.66e3i)T^{2} \)
17 \( 1 + (-73.7 - 47.3i)T + (2.04e3 + 4.46e3i)T^{2} \)
19 \( 1 + (-53.8 + 34.5i)T + (2.84e3 - 6.23e3i)T^{2} \)
29 \( 1 + (192. + 123. i)T + (1.01e4 + 2.21e4i)T^{2} \)
31 \( 1 + (115. - 133. i)T + (-4.23e3 - 2.94e4i)T^{2} \)
37 \( 1 + (51.7 - 360. i)T + (-4.86e4 - 1.42e4i)T^{2} \)
41 \( 1 + (-30.4 - 211. i)T + (-6.61e4 + 1.94e4i)T^{2} \)
43 \( 1 + (-301. - 347. i)T + (-1.13e4 + 7.86e4i)T^{2} \)
47 \( 1 - 78.1T + 1.03e5T^{2} \)
53 \( 1 + (138. - 302. i)T + (-9.74e4 - 1.12e5i)T^{2} \)
59 \( 1 + (-150. - 328. i)T + (-1.34e5 + 1.55e5i)T^{2} \)
61 \( 1 + (-74.6 + 86.1i)T + (-3.23e4 - 2.24e5i)T^{2} \)
67 \( 1 + (830. - 243. i)T + (2.53e5 - 1.62e5i)T^{2} \)
71 \( 1 + (570. - 167. i)T + (3.01e5 - 1.93e5i)T^{2} \)
73 \( 1 + (-374. + 240. i)T + (1.61e5 - 3.53e5i)T^{2} \)
79 \( 1 + (428. + 938. i)T + (-3.22e5 + 3.72e5i)T^{2} \)
83 \( 1 + (38.6 - 268. i)T + (-5.48e5 - 1.61e5i)T^{2} \)
89 \( 1 + (317. + 366. i)T + (-1.00e5 + 6.97e5i)T^{2} \)
97 \( 1 + (-70.0 - 487. i)T + (-8.75e5 + 2.57e5i)T^{2} \)
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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

−12.60111290064165210688025770751, −11.52546797171514751200008795814, −10.64463099502918093411988363227, −10.03694579959020730891363050212, −7.75950235963604100619682240708, −7.11358427130490988651412724672, −5.98247202873278119603552301946, −4.50385168232413198041289265789, −3.13752616622077965575691061895, −1.22808593315392066964584530063, 1.79301815393198197402557612599, 3.89951204471479512745127089377, 5.27291905081496964334235258993, 5.65935215198520826062761438046, 7.40414091301860067714010353038, 9.029703364542256005438237620567, 9.319189519419717807955540490427, 11.38114071941655976997796417735, 11.94034429772913110799502971805, 12.55435343405174171519345342131

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