↑

Properties

Label 2-138-23.22-c4-0-8
Degree $2$
Conductor $138$
Sign $0.841 - 0.539i$
Analytic cond. $14.2650$
Root an. cond. $3.77691$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2.82·2-s + 5.19·3-s + 8.00·4-s − 7.70i·5-s + 14.6·6-s + 72.1i·7-s + 22.6·8-s + 27·9-s − 21.7i·10-s + 99.6i·11-s + 41.5·12-s + 228.·13-s + 203. i·14-s − 40.0i·15-s + 64.0·16-s + 208. i·17-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 0.500·4-s − 0.308i·5-s + 0.408·6-s + 1.47i·7-s + 0.353·8-s + 0.333·9-s − 0.217i·10-s + 0.823i·11-s + 0.288·12-s + 1.35·13-s + 1.04i·14-s − 0.177i·15-s + 0.250·16-s + 0.722i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.841 - 0.539i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.841 - 0.539i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(138\)    =    \(2 \cdot 3 \cdot 23\)
Sign: $0.841 - 0.539i$
Analytic conductor: \(14.2650\)
Root analytic conductor: \(3.77691\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{138} (91, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 138,\ (\ :2),\ 0.841 - 0.539i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(3.34889 + 0.981015i\)
\(L(\frac12)\) \(\approx\) \(3.34889 + 0.981015i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 2.82T \)
3 \( 1 - 5.19T \)
23 \( 1 + (-445. + 285. i)T \)
good5 \( 1 + 7.70iT - 625T^{2} \)
7 \( 1 - 72.1iT - 2.40e3T^{2} \)
11 \( 1 - 99.6iT - 1.46e4T^{2} \)
13 \( 1 - 228.T + 2.85e4T^{2} \)
17 \( 1 - 208. iT - 8.35e4T^{2} \)
19 \( 1 + 517. iT - 1.30e5T^{2} \)
29 \( 1 + 1.44e3T + 7.07e5T^{2} \)
31 \( 1 + 578.T + 9.23e5T^{2} \)
37 \( 1 - 1.88e3iT - 1.87e6T^{2} \)
41 \( 1 + 302.T + 2.82e6T^{2} \)
43 \( 1 + 2.30e3iT - 3.41e6T^{2} \)
47 \( 1 - 1.28e3T + 4.87e6T^{2} \)
53 \( 1 + 979. iT - 7.89e6T^{2} \)
59 \( 1 + 3.43e3T + 1.21e7T^{2} \)
61 \( 1 + 2.67e3iT - 1.38e7T^{2} \)
67 \( 1 - 6.72e3iT - 2.01e7T^{2} \)
71 \( 1 + 8.23e3T + 2.54e7T^{2} \)
73 \( 1 - 468.T + 2.83e7T^{2} \)
79 \( 1 + 1.12e4iT - 3.89e7T^{2} \)
83 \( 1 + 1.12e4iT - 4.74e7T^{2} \)
89 \( 1 + 9.35e3iT - 6.27e7T^{2} \)
97 \( 1 - 6.82e3iT - 8.85e7T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.85465413213245383003477232476, −11.78450816181947297160036183874, −10.72699310800811781064805388716, −9.132285588945344054903924361733, −8.595956281631946348393058459568, −7.05037006770146543267656253392, −5.80932574449438432040838441169, −4.64539497941799894095463464792, −3.14003991922338209795012061388, −1.83426960345629751181697801682, 1.24905621008589371217489535835, 3.29651437801813220141189609352, 4.00558134197853862795365426643, 5.72437570069158783377719839588, 7.01411376981608105812920972777, 7.893618531258269448474710942894, 9.282165967506839948829990995849, 10.72452389057534919854406290617, 11.12496236375092360343942288437, 12.75741533268682593524974944201

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