↑

Properties

Label 2-363-33.20-c2-0-30
Degree $2$
Conductor $363$
Sign $0.241 - 0.970i$
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  + (3.64 + 1.18i)2-s + (−2.84 − 0.950i)3-s + (8.65 + 6.28i)4-s + (−0.590 + 0.191i)5-s + (−9.24 − 6.83i)6-s + (1.55 + 1.13i)7-s + (15.0 + 20.7i)8-s + (7.19 + 5.40i)9-s − 2.37·10-s + (−18.6 − 26.1i)12-s + (−4.92 + 15.1i)13-s + (4.34 + 5.97i)14-s + (1.86 + 0.0152i)15-s + (17.1 + 52.8i)16-s + (13.7 − 4.46i)17-s + (19.8 + 28.2i)18-s + ⋯
L(s)  = 1  + (1.82 + 0.592i)2-s + (−0.948 − 0.316i)3-s + (2.16 + 1.57i)4-s + (−0.118 + 0.0383i)5-s + (−1.54 − 1.13i)6-s + (0.222 + 0.161i)7-s + (1.88 + 2.59i)8-s + (0.799 + 0.600i)9-s − 0.237·10-s + (−1.55 − 2.17i)12-s + (−0.379 + 1.16i)13-s + (0.310 + 0.426i)14-s + (0.124 + 0.00101i)15-s + (1.07 + 3.30i)16-s + (0.807 − 0.262i)17-s + (1.10 + 1.56i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.241 - 0.970i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.241 - 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(363\)    =    \(3 \cdot 11^{2}\)
Sign: $0.241 - 0.970i$
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: \((2,\ 363,\ (\ :1),\ 0.241 - 0.970i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.98626 + 2.33424i\)
\(L(\frac12)\) \(\approx\) \(2.98626 + 2.33424i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (2.84 + 0.950i)T \)
11 \( 1 \)
good2 \( 1 + (-3.64 - 1.18i)T + (3.23 + 2.35i)T^{2} \)
5 \( 1 + (0.590 - 0.191i)T + (20.2 - 14.6i)T^{2} \)
7 \( 1 + (-1.55 - 1.13i)T + (15.1 + 46.6i)T^{2} \)
13 \( 1 + (4.92 - 15.1i)T + (-136. - 99.3i)T^{2} \)
17 \( 1 + (-13.7 + 4.46i)T + (233. - 169. i)T^{2} \)
19 \( 1 + (-13.3 + 9.69i)T + (111. - 343. i)T^{2} \)
23 \( 1 - 10.6iT - 529T^{2} \)
29 \( 1 + (-11.2 + 15.5i)T + (-259. - 799. i)T^{2} \)
31 \( 1 + (-6.80 + 20.9i)T + (-777. - 564. i)T^{2} \)
37 \( 1 + (-16.7 - 12.1i)T + (423. + 1.30e3i)T^{2} \)
41 \( 1 + (15.8 + 21.8i)T + (-519. + 1.59e3i)T^{2} \)
43 \( 1 + 53.1T + 1.84e3T^{2} \)
47 \( 1 + (26.8 + 36.9i)T + (-682. + 2.10e3i)T^{2} \)
53 \( 1 + (-57.4 - 18.6i)T + (2.27e3 + 1.65e3i)T^{2} \)
59 \( 1 + (-23.8 + 32.7i)T + (-1.07e3 - 3.31e3i)T^{2} \)
61 \( 1 + (19.8 + 60.9i)T + (-3.01e3 + 2.18e3i)T^{2} \)
67 \( 1 + 25.7T + 4.48e3T^{2} \)
71 \( 1 + (-108. + 35.2i)T + (4.07e3 - 2.96e3i)T^{2} \)
73 \( 1 + (68.3 + 49.6i)T + (1.64e3 + 5.06e3i)T^{2} \)
79 \( 1 + (-20.1 + 62.0i)T + (-5.04e3 - 3.66e3i)T^{2} \)
83 \( 1 + (-85.2 + 27.7i)T + (5.57e3 - 4.04e3i)T^{2} \)
89 \( 1 - 146. iT - 7.92e3T^{2} \)
97 \( 1 + (-19.2 + 59.3i)T + (-7.61e3 - 5.53e3i)T^{2} \)
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

−11.80547063194876690654746200043, −11.20799782672007416009487074601, −9.793668349558232949879260946986, −7.956443938688181536027130803082, −7.16609046383982628638827163907, −6.40954038725426049381397901935, −5.40545764816313977377985063645, −4.76387910254775463044903738723, −3.61481454360006171293631386712, −2.01887298678632333957787000921, 1.20735561688962345911725035096, 3.01465967405955978775813205191, 4.06133097994749488291175179685, 5.07405830712625003470595907680, 5.68874350290893476270688576993, 6.65817963024098204019396646630, 7.80498126767160229042564757002, 9.985081108979150296180300871525, 10.37145019435365518731634842896, 11.33099340785612128393637479308

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