↑

Properties

Label 2-133-133.102-c1-0-1
Degree $2$
Conductor $133$
Sign $-0.0592 - 0.998i$
Analytic cond. $1.06201$
Root an. cond. $1.03053$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.433 − 0.751i)2-s − 2.55·3-s + (0.623 + 1.07i)4-s + (−1.78 + 3.08i)5-s + (−1.11 + 1.92i)6-s + (−2.63 − 0.185i)7-s + 2.81·8-s + 3.55·9-s + (1.54 + 2.67i)10-s + (−1.71 + 2.96i)11-s + (−1.59 − 2.76i)12-s + (1.01 − 1.76i)13-s + (−1.28 + 1.90i)14-s + (4.56 − 7.90i)15-s + (−0.0237 + 0.0411i)16-s + 5.19·17-s + ⋯
L(s)  = 1  + (0.306 − 0.531i)2-s − 1.47·3-s + (0.311 + 0.539i)4-s + (−0.797 + 1.38i)5-s + (−0.453 + 0.785i)6-s + (−0.997 − 0.0703i)7-s + 0.996·8-s + 1.18·9-s + (0.489 + 0.847i)10-s + (−0.516 + 0.895i)11-s + (−0.460 − 0.797i)12-s + (0.282 − 0.488i)13-s + (−0.343 + 0.508i)14-s + (1.17 − 2.03i)15-s + (−0.00593 + 0.0102i)16-s + 1.26·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 133 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0592 - 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 133 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0592 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(133\)    =    \(7 \cdot 19\)
Sign: $-0.0592 - 0.998i$
Analytic conductor: \(1.06201\)
Root analytic conductor: \(1.03053\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{133} (102, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 133,\ (\ :1/2),\ -0.0592 - 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.402623 + 0.427234i\)
\(L(\frac12)\) \(\approx\) \(0.402623 + 0.427234i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (2.63 + 0.185i)T \)
19 \( 1 + (4.00 - 1.72i)T \)
good2 \( 1 + (-0.433 + 0.751i)T + (-1 - 1.73i)T^{2} \)
3 \( 1 + 2.55T + 3T^{2} \)
5 \( 1 + (1.78 - 3.08i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (1.71 - 2.96i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-1.01 + 1.76i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 5.19T + 17T^{2} \)
23 \( 1 + 0.934T + 23T^{2} \)
29 \( 1 + (1.90 - 3.29i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-0.776 + 1.34i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (2.32 + 4.02i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-5.71 - 9.90i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-1.90 - 3.30i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 10.4T + 47T^{2} \)
53 \( 1 + (0.138 + 0.240i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 - 0.461T + 59T^{2} \)
61 \( 1 - 2.59T + 61T^{2} \)
67 \( 1 + (-2.18 - 3.77i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (1.39 + 2.42i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + 13.2T + 73T^{2} \)
79 \( 1 + (-2.69 + 4.67i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 15.6T + 83T^{2} \)
89 \( 1 + 1.16T + 89T^{2} \)
97 \( 1 + (1.27 + 2.20i)T + (-48.5 + 84.0i)T^{2} \)
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

−12.91198227538853733093853724143, −12.37647453283861204654543491267, −11.52053918469907642611755293951, −10.62780711662962153227783763209, −10.18926613486675689425591342881, −7.71261158838312075780298985126, −6.97123303867836810679132031999, −5.90790304390216695201733209084, −4.13460849109350230795312697996, −2.93948774768379457094955880261, 0.65609821746186950731705060034, 4.21319275190999660537966214264, 5.44400385237555004523349872598, 6.00422359203235824234097600349, 7.26974309832662756924884212596, 8.700254993061413860302139180534, 10.14436439333809411136639526418, 11.10380127395499405828693366638, 12.06573148776266805687386009259, 12.78458335295622638133837735209

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