Properties

Label 2-138-3.2-c2-0-7
Degree $2$
Conductor $138$
Sign $0.999 - 0.0408i$
Analytic cond. $3.76022$
Root an. cond. $1.93913$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.41i·2-s + (−2.99 + 0.122i)3-s − 2.00·4-s − 4.36i·5-s + (−0.173 − 4.23i)6-s + 5.99·7-s − 2.82i·8-s + (8.96 − 0.734i)9-s + 6.17·10-s − 0.167i·11-s + (5.99 − 0.245i)12-s + 15.4·13-s + 8.48i·14-s + (0.535 + 13.0i)15-s + 4.00·16-s − 12.8i·17-s + ⋯
L(s)  = 1  + 0.707i·2-s + (−0.999 + 0.0408i)3-s − 0.500·4-s − 0.873i·5-s + (−0.0288 − 0.706i)6-s + 0.857·7-s − 0.353i·8-s + (0.996 − 0.0816i)9-s + 0.617·10-s − 0.0151i·11-s + (0.499 − 0.0204i)12-s + 1.18·13-s + 0.606i·14-s + (0.0356 + 0.872i)15-s + 0.250·16-s − 0.754i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0408i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.999 - 0.0408i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(138\)    =    \(2 \cdot 3 \cdot 23\)
Sign: $0.999 - 0.0408i$
Analytic conductor: \(3.76022\)
Root analytic conductor: \(1.93913\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{138} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 138,\ (\ :1),\ 0.999 - 0.0408i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.10255 + 0.0225354i\)
\(L(\frac12)\) \(\approx\) \(1.10255 + 0.0225354i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 1.41iT \)
3 \( 1 + (2.99 - 0.122i)T \)
23 \( 1 + 4.79iT \)
good5 \( 1 + 4.36iT - 25T^{2} \)
7 \( 1 - 5.99T + 49T^{2} \)
11 \( 1 + 0.167iT - 121T^{2} \)
13 \( 1 - 15.4T + 169T^{2} \)
17 \( 1 + 12.8iT - 289T^{2} \)
19 \( 1 - 11.3T + 361T^{2} \)
29 \( 1 + 52.1iT - 841T^{2} \)
31 \( 1 + 1.17T + 961T^{2} \)
37 \( 1 + 7.85T + 1.36e3T^{2} \)
41 \( 1 - 25.1iT - 1.68e3T^{2} \)
43 \( 1 - 24.9T + 1.84e3T^{2} \)
47 \( 1 - 40.5iT - 2.20e3T^{2} \)
53 \( 1 + 73.1iT - 2.80e3T^{2} \)
59 \( 1 - 79.5iT - 3.48e3T^{2} \)
61 \( 1 - 17.5T + 3.72e3T^{2} \)
67 \( 1 + 113.T + 4.48e3T^{2} \)
71 \( 1 - 117. iT - 5.04e3T^{2} \)
73 \( 1 + 41.4T + 5.32e3T^{2} \)
79 \( 1 - 134.T + 6.24e3T^{2} \)
83 \( 1 + 22.9iT - 6.88e3T^{2} \)
89 \( 1 - 175. iT - 7.92e3T^{2} \)
97 \( 1 + 26.9T + 9.40e3T^{2} \)
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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

−13.02528865242060650127152282897, −11.88875096858505049769118112084, −11.10435234241465706777861911148, −9.760876346852509358443689191456, −8.611258656420284163816198141312, −7.56702837206556168668004022715, −6.20388439853306266975194940567, −5.20670503263239289050962658180, −4.28922566122352442971704194151, −1.02472747462430261215618604309, 1.49627556011356977178597839658, 3.56584997258296993694122501731, 5.02083497364782468385001330318, 6.24489383557662929933920493460, 7.51305245290660108175468533222, 8.912090282805320065850977820080, 10.48787052047971699926814505976, 10.84348624743057425423397172925, 11.69389143163223459322582384614, 12.69563069799612920847692962226

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