↑

Properties

Label 2-138-23.22-c4-0-13
Degree $2$
Conductor $138$
Sign $-0.998 + 0.0541i$
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 − 30.1i·5-s + 14.6·6-s − 56.6i·7-s − 22.6·8-s + 27·9-s + 85.3i·10-s + 36.1i·11-s − 41.5·12-s + 133.·13-s + 160. i·14-s + 156. i·15-s + 64.0·16-s − 66.1i·17-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 0.500·4-s − 1.20i·5-s + 0.408·6-s − 1.15i·7-s − 0.353·8-s + 0.333·9-s + 0.853i·10-s + 0.298i·11-s − 0.288·12-s + 0.789·13-s + 0.818i·14-s + 0.697i·15-s + 0.250·16-s − 0.228i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 + 0.0541i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.998 + 0.0541i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(138\)    =    \(2 \cdot 3 \cdot 23\)
Sign: $-0.998 + 0.0541i$
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: \((2,\ 138,\ (\ :2),\ -0.998 + 0.0541i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.0148856 - 0.549841i\)
\(L(\frac12)\) \(\approx\) \(0.0148856 - 0.549841i\)
\(L(3)\) 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 \( 1 + (528. - 28.6i)T \)
good5 \( 1 + 30.1iT - 625T^{2} \)
7 \( 1 + 56.6iT - 2.40e3T^{2} \)
11 \( 1 - 36.1iT - 1.46e4T^{2} \)
13 \( 1 - 133.T + 2.85e4T^{2} \)
17 \( 1 + 66.1iT - 8.35e4T^{2} \)
19 \( 1 + 298. iT - 1.30e5T^{2} \)
29 \( 1 + 913.T + 7.07e5T^{2} \)
31 \( 1 + 571.T + 9.23e5T^{2} \)
37 \( 1 - 821. iT - 1.87e6T^{2} \)
41 \( 1 + 1.67e3T + 2.82e6T^{2} \)
43 \( 1 + 395. iT - 3.41e6T^{2} \)
47 \( 1 + 1.09e3T + 4.87e6T^{2} \)
53 \( 1 + 2.75e3iT - 7.89e6T^{2} \)
59 \( 1 + 1.07e3T + 1.21e7T^{2} \)
61 \( 1 - 5.88e3iT - 1.38e7T^{2} \)
67 \( 1 + 2.37e3iT - 2.01e7T^{2} \)
71 \( 1 + 3.27e3T + 2.54e7T^{2} \)
73 \( 1 - 6.20e3T + 2.83e7T^{2} \)
79 \( 1 + 426. iT - 3.89e7T^{2} \)
83 \( 1 - 1.60e3iT - 4.74e7T^{2} \)
89 \( 1 - 1.36e4iT - 6.27e7T^{2} \)
97 \( 1 - 9.81e3iT - 8.85e7T^{2} \)
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

−11.82833869071428839388071383616, −10.89544138240148267987278182609, −9.904300399246238280689016985444, −8.883011590885236682566861928283, −7.76894080629097216583939197677, −6.66691944750064032763242344852, −5.23687965304948264595993511514, −3.98018475129137012250476605443, −1.46116192743396777635584728906, −0.31199494381736314259877208393, 1.95386461946138755303533298285, 3.47436357965158825487236540536, 5.71278281129170377118162152967, 6.39466842132799805988811621270, 7.68826531278210074173112118946, 8.846792553138662028062826692075, 10.03906381387373653184412610446, 10.94138587051392167151043067625, 11.65370533296786269371082579957, 12.68489748737781823047674574446

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