↑

Properties

Label 2-138-69.68-c5-0-19
Degree $2$
Conductor $138$
Sign $0.844 - 0.535i$
Analytic cond. $22.1329$
Root an. cond. $4.70456$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 4i·2-s + (14.2 − 6.20i)3-s − 16·4-s − 97.9·5-s + (24.8 + 57.1i)6-s + 18.8i·7-s − 64i·8-s + (165. − 177. i)9-s − 391. i·10-s − 79.1·11-s + (−228. + 99.3i)12-s + 704.·13-s − 75.3·14-s + (−1.40e3 + 608. i)15-s + 256·16-s + 2.09e3·17-s + ⋯
L(s)  = 1  + 0.707i·2-s + (0.917 − 0.398i)3-s − 0.5·4-s − 1.75·5-s + (0.281 + 0.648i)6-s + 0.145i·7-s − 0.353i·8-s + (0.682 − 0.730i)9-s − 1.23i·10-s − 0.197·11-s + (−0.458 + 0.199i)12-s + 1.15·13-s − 0.102·14-s + (−1.60 + 0.697i)15-s + 0.250·16-s + 1.76·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.844 - 0.535i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.844 - 0.535i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(138\)    =    \(2 \cdot 3 \cdot 23\)
Sign: $0.844 - 0.535i$
Analytic conductor: \(22.1329\)
Root analytic conductor: \(4.70456\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{138} (137, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 138,\ (\ :5/2),\ 0.844 - 0.535i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.929857629\)
\(L(\frac12)\) \(\approx\) \(1.929857629\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 4iT \)
3 \( 1 + (-14.2 + 6.20i)T \)
23 \( 1 + (-392. - 2.50e3i)T \)
good5 \( 1 + 97.9T + 3.12e3T^{2} \)
7 \( 1 - 18.8iT - 1.68e4T^{2} \)
11 \( 1 + 79.1T + 1.61e5T^{2} \)
13 \( 1 - 704.T + 3.71e5T^{2} \)
17 \( 1 - 2.09e3T + 1.41e6T^{2} \)
19 \( 1 - 1.64e3iT - 2.47e6T^{2} \)
29 \( 1 + 7.95e3iT - 2.05e7T^{2} \)
31 \( 1 - 2.88e3T + 2.86e7T^{2} \)
37 \( 1 + 3.44e3iT - 6.93e7T^{2} \)
41 \( 1 - 1.22e4iT - 1.15e8T^{2} \)
43 \( 1 + 2.11e4iT - 1.47e8T^{2} \)
47 \( 1 - 2.01e4iT - 2.29e8T^{2} \)
53 \( 1 - 1.08e4T + 4.18e8T^{2} \)
59 \( 1 + 5.68e3iT - 7.14e8T^{2} \)
61 \( 1 - 3.99e4iT - 8.44e8T^{2} \)
67 \( 1 + 2.83e4iT - 1.35e9T^{2} \)
71 \( 1 - 6.11e3iT - 1.80e9T^{2} \)
73 \( 1 + 1.99e4T + 2.07e9T^{2} \)
79 \( 1 - 4.22e3iT - 3.07e9T^{2} \)
83 \( 1 + 4.33e4T + 3.93e9T^{2} \)
89 \( 1 - 8.98e4T + 5.58e9T^{2} \)
97 \( 1 + 1.04e5iT - 8.58e9T^{2} \)
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

−12.39078059730794961847163152976, −11.65675869810517038930735008341, −10.08094022830453547391547464077, −8.724232793094963102919299223362, −7.890181519952482207292018235320, −7.48479805168911635157028266940, −5.93656710223270855586787006832, −4.08564446587622860742431411989, −3.34233277444636995111210955919, −0.946647988551374564193777287912, 0.926197864585291640443980126452, 3.04771401161699384117774191913, 3.76091533793709771640137575993, 4.87215792074572090940814837009, 7.20118835641970124058932840143, 8.216080424011094563659120676094, 8.855730990427017442137928530501, 10.31835097285829553688164737353, 11.07842339441688443628295438480, 12.13205209030244922595987552398

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