↑

Properties

Label 2-133-133.102-c1-0-10
Degree $2$
Conductor $133$
Sign $0.157 + 0.987i$
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  + (1.14 − 1.98i)2-s + 2.15·3-s + (−1.62 − 2.81i)4-s + (−1.58 + 2.74i)5-s + (2.46 − 4.27i)6-s + (−2.63 − 0.218i)7-s − 2.87·8-s + 1.63·9-s + (3.63 + 6.30i)10-s + (1.48 − 2.56i)11-s + (−3.50 − 6.06i)12-s + (−2.75 + 4.77i)13-s + (−3.45 + 4.98i)14-s + (−3.41 + 5.91i)15-s + (−0.0401 + 0.0694i)16-s − 0.323·17-s + ⋯
L(s)  = 1  + (0.810 − 1.40i)2-s + 1.24·3-s + (−0.813 − 1.40i)4-s + (−0.709 + 1.22i)5-s + (1.00 − 1.74i)6-s + (−0.996 − 0.0826i)7-s − 1.01·8-s + 0.544·9-s + (1.15 + 1.99i)10-s + (0.447 − 0.774i)11-s + (−1.01 − 1.75i)12-s + (−0.764 + 1.32i)13-s + (−0.923 + 1.33i)14-s + (−0.882 + 1.52i)15-s + (−0.0100 + 0.0173i)16-s − 0.0783·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 133 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.157 + 0.987i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 133 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.157 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(133\)    =    \(7 \cdot 19\)
Sign: $0.157 + 0.987i$
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: \((2,\ 133,\ (\ :1/2),\ 0.157 + 0.987i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.38316 - 1.18020i\)
\(L(\frac12)\) \(\approx\) \(1.38316 - 1.18020i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (2.63 + 0.218i)T \)
19 \( 1 + (-3.78 + 2.15i)T \)
good2 \( 1 + (-1.14 + 1.98i)T + (-1 - 1.73i)T^{2} \)
3 \( 1 - 2.15T + 3T^{2} \)
5 \( 1 + (1.58 - 2.74i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-1.48 + 2.56i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (2.75 - 4.77i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 0.323T + 17T^{2} \)
23 \( 1 + 2.19T + 23T^{2} \)
29 \( 1 + (-0.399 + 0.692i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-4.58 + 7.93i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-4.67 - 8.10i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-0.864 - 1.49i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-1.27 - 2.20i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 3.05T + 47T^{2} \)
53 \( 1 + (2.50 + 4.33i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + 7.00T + 59T^{2} \)
61 \( 1 - 2.47T + 61T^{2} \)
67 \( 1 + (-3.75 - 6.50i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (5.16 + 8.94i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + 0.510T + 73T^{2} \)
79 \( 1 + (2.77 - 4.81i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 9.91T + 83T^{2} \)
89 \( 1 + 17.1T + 89T^{2} \)
97 \( 1 + (-1.59 - 2.75i)T + (-48.5 + 84.0i)T^{2} \)
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

−13.22626661414413391614800512424, −11.77979053284450253449295782095, −11.37597367797936131143015374004, −9.982033864686337053384127477007, −9.315966247330602512172239788757, −7.70952354371193452723525876612, −6.45514874645602407845873704982, −4.19458626302991231138632383972, −3.26756490988806951282528327216, −2.56527072819520852397380938845, 3.29014176431283196285894937865, 4.44774498464579066867153843459, 5.65540526659393130640425652547, 7.24057041669489120582091012093, 7.969302698832474246905996283265, 8.854458982318878762611546370170, 9.876396619396521541886884928659, 12.38261236461722878282324276593, 12.63270079146331306979851626863, 13.72353787547141999671587980913

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