↑

Properties

Label 2-220-55.17-c1-0-5
Degree $2$
Conductor $220$
Sign $0.0835 + 0.996i$
Analytic cond. $1.75670$
Root an. cond. $1.32540$
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  + (−0.464 + 0.0735i)3-s + (−0.794 − 2.09i)5-s + (0.413 − 2.61i)7-s + (−2.64 + 0.858i)9-s + (1.70 − 2.84i)11-s + (0.780 − 1.53i)13-s + (0.522 + 0.911i)15-s + (−1.07 − 2.11i)17-s + (2.28 + 1.65i)19-s + 1.24i·21-s + (4.09 − 4.09i)23-s + (−3.73 + 3.31i)25-s + (2.41 − 1.23i)27-s + (−1.79 + 1.30i)29-s + (1.63 + 5.02i)31-s + ⋯
L(s)  = 1  + (−0.267 + 0.0424i)3-s + (−0.355 − 0.934i)5-s + (0.156 − 0.987i)7-s + (−0.881 + 0.286i)9-s + (0.514 − 0.857i)11-s + (0.216 − 0.425i)13-s + (0.134 + 0.235i)15-s + (−0.261 − 0.512i)17-s + (0.523 + 0.380i)19-s + 0.271i·21-s + (0.854 − 0.854i)23-s + (−0.747 + 0.663i)25-s + (0.465 − 0.237i)27-s + (−0.333 + 0.242i)29-s + (0.293 + 0.902i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 220 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0835 + 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 220 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0835 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(220\)    =    \(2^{2} \cdot 5 \cdot 11\)
Sign: $0.0835 + 0.996i$
Analytic conductor: \(1.75670\)
Root analytic conductor: \(1.32540\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{220} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 220,\ (\ :1/2),\ 0.0835 + 0.996i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.703051 - 0.646564i\)
\(L(\frac12)\) \(\approx\) \(0.703051 - 0.646564i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (0.794 + 2.09i)T \)
11 \( 1 + (-1.70 + 2.84i)T \)
good3 \( 1 + (0.464 - 0.0735i)T + (2.85 - 0.927i)T^{2} \)
7 \( 1 + (-0.413 + 2.61i)T + (-6.65 - 2.16i)T^{2} \)
13 \( 1 + (-0.780 + 1.53i)T + (-7.64 - 10.5i)T^{2} \)
17 \( 1 + (1.07 + 2.11i)T + (-9.99 + 13.7i)T^{2} \)
19 \( 1 + (-2.28 - 1.65i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + (-4.09 + 4.09i)T - 23iT^{2} \)
29 \( 1 + (1.79 - 1.30i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (-1.63 - 5.02i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (2.56 + 0.405i)T + (35.1 + 11.4i)T^{2} \)
41 \( 1 + (6.37 - 8.77i)T + (-12.6 - 38.9i)T^{2} \)
43 \( 1 + (-4.89 - 4.89i)T + 43iT^{2} \)
47 \( 1 + (1.64 + 10.3i)T + (-44.6 + 14.5i)T^{2} \)
53 \( 1 + (-8.77 - 4.47i)T + (31.1 + 42.8i)T^{2} \)
59 \( 1 + (2.11 + 2.91i)T + (-18.2 + 56.1i)T^{2} \)
61 \( 1 + (-11.5 - 3.74i)T + (49.3 + 35.8i)T^{2} \)
67 \( 1 + (-4.30 - 4.30i)T + 67iT^{2} \)
71 \( 1 + (-0.522 + 1.60i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (14.3 + 2.26i)T + (69.4 + 22.5i)T^{2} \)
79 \( 1 + (-2.50 - 7.70i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (-8.02 + 4.08i)T + (48.7 - 67.1i)T^{2} \)
89 \( 1 + 11.6iT - 89T^{2} \)
97 \( 1 + (-4.57 + 8.98i)T + (-57.0 - 78.4i)T^{2} \)
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

−11.88030139421175968245246995455, −11.22726323914039347087210890372, −10.26697157388226112337651021005, −8.878224108543834864043430315927, −8.256784215194765479596499415808, −7.02579269026141870343419094597, −5.66343233663628813772758264111, −4.66130832598507651984754492008, −3.33225061927011388659344795072, −0.870204952360649524822876012924, 2.34687084713542113902114177822, 3.74332503405092731026501853295, 5.35524377818991351005875694975, 6.39850607150078339014216549156, 7.35730665364908957588931754895, 8.671483962914801097433431428057, 9.510950576869478702914700234533, 10.82098477091492432930647260677, 11.66258090136669461476022735014, 12.11577928268090652344505823893

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