↑

Properties

Label 2-363-33.20-c2-0-47
Degree $2$
Conductor $363$
Sign $0.369 - 0.929i$
Analytic cond. $9.89103$
Root an. cond. $3.14500$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (2.91 + 0.945i)2-s + (1.65 + 2.49i)3-s + (4.34 + 3.15i)4-s + (6.31 − 2.05i)5-s + (2.46 + 8.84i)6-s + (−2.47 − 1.80i)7-s + (2.45 + 3.37i)8-s + (−3.49 + 8.29i)9-s + 20.3·10-s + (−0.678 + 16.0i)12-s + (−5.01 + 15.4i)13-s + (−5.50 − 7.58i)14-s + (15.6 + 12.3i)15-s + (−2.68 − 8.25i)16-s + (0.766 − 0.248i)17-s + (−18.0 + 20.8i)18-s + ⋯
L(s)  = 1  + (1.45 + 0.472i)2-s + (0.553 + 0.833i)3-s + (1.08 + 0.788i)4-s + (1.26 − 0.410i)5-s + (0.411 + 1.47i)6-s + (−0.353 − 0.257i)7-s + (0.306 + 0.422i)8-s + (−0.388 + 0.921i)9-s + 2.03·10-s + (−0.0565 + 1.34i)12-s + (−0.386 + 1.18i)13-s + (−0.393 − 0.541i)14-s + (1.04 + 0.825i)15-s + (−0.167 − 0.515i)16-s + (0.0450 − 0.0146i)17-s + (−1.00 + 1.15i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.369 - 0.929i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.369 - 0.929i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(363\)    =    \(3 \cdot 11^{2}\)
Sign: $0.369 - 0.929i$
Analytic conductor: \(9.89103\)
Root analytic conductor: \(3.14500\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{363} (251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 363,\ (\ :1),\ 0.369 - 0.929i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(3.98676 + 2.70558i\)
\(L(\frac12)\) \(\approx\) \(3.98676 + 2.70558i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.65 - 2.49i)T \)
11 \( 1 \)
good2 \( 1 + (-2.91 - 0.945i)T + (3.23 + 2.35i)T^{2} \)
5 \( 1 + (-6.31 + 2.05i)T + (20.2 - 14.6i)T^{2} \)
7 \( 1 + (2.47 + 1.80i)T + (15.1 + 46.6i)T^{2} \)
13 \( 1 + (5.01 - 15.4i)T + (-136. - 99.3i)T^{2} \)
17 \( 1 + (-0.766 + 0.248i)T + (233. - 169. i)T^{2} \)
19 \( 1 + (-16.7 + 12.1i)T + (111. - 343. i)T^{2} \)
23 \( 1 + 27.3iT - 529T^{2} \)
29 \( 1 + (2.22 - 3.06i)T + (-259. - 799. i)T^{2} \)
31 \( 1 + (6.42 - 19.7i)T + (-777. - 564. i)T^{2} \)
37 \( 1 + (31.1 + 22.6i)T + (423. + 1.30e3i)T^{2} \)
41 \( 1 + (-7.86 - 10.8i)T + (-519. + 1.59e3i)T^{2} \)
43 \( 1 + 43.4T + 1.84e3T^{2} \)
47 \( 1 + (-11.6 - 16.0i)T + (-682. + 2.10e3i)T^{2} \)
53 \( 1 + (-16.8 - 5.46i)T + (2.27e3 + 1.65e3i)T^{2} \)
59 \( 1 + (25.5 - 35.2i)T + (-1.07e3 - 3.31e3i)T^{2} \)
61 \( 1 + (3.29 + 10.1i)T + (-3.01e3 + 2.18e3i)T^{2} \)
67 \( 1 - 72.2T + 4.48e3T^{2} \)
71 \( 1 + (-2.44 + 0.794i)T + (4.07e3 - 2.96e3i)T^{2} \)
73 \( 1 + (36.7 + 26.7i)T + (1.64e3 + 5.06e3i)T^{2} \)
79 \( 1 + (-30.3 + 93.4i)T + (-5.04e3 - 3.66e3i)T^{2} \)
83 \( 1 + (-30.3 + 9.85i)T + (5.57e3 - 4.04e3i)T^{2} \)
89 \( 1 + 18.5iT - 7.92e3T^{2} \)
97 \( 1 + (19.5 - 60.2i)T + (-7.61e3 - 5.53e3i)T^{2} \)
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

−11.60433990796774898275270956800, −10.34192105525963205986024111329, −9.506658970238243758109742330665, −8.848446442437751514102349688014, −7.22847645823687222776316415754, −6.29840740997998707814257752569, −5.21908874807811316758014002342, −4.62289473563733649568022852717, −3.43959453614459363752368363883, −2.23985211080503827597293540726, 1.70814922313784356994346333417, 2.76298966005525945584519974841, 3.50568354467449297758221545947, 5.39807299160904814418887079001, 5.85741927625649039572371144941, 6.88473026912828984978983250708, 8.069967274037474522957282147645, 9.431611958613297022480516470178, 10.17776763649620875097398407588, 11.41597361214923872271533954344

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