↑

Properties

Label 2-363-33.17-c1-0-8
Degree $2$
Conductor $363$
Sign $0.906 + 0.421i$
Analytic cond. $2.89856$
Root an. cond. $1.70251$
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  + (−1.40 − 1.01i)3-s + (1.61 − 1.17i)4-s + (2.79 + 3.85i)7-s + (0.927 + 2.85i)9-s − 3.46·12-s + (3.31 − 1.07i)13-s + (1.23 − 3.80i)16-s + (0.749 − 1.03i)19-s − 8.24i·21-s + (−4.04 − 2.93i)25-s + (1.60 − 4.94i)27-s + (9.05 + 2.94i)28-s + (−0.535 − 1.64i)31-s + (4.85 + 3.52i)36-s + (4.20 − 3.05i)37-s + ⋯
L(s)  = 1  + (−0.809 − 0.587i)3-s + (0.809 − 0.587i)4-s + (1.05 + 1.45i)7-s + (0.309 + 0.951i)9-s − 1.00·12-s + (0.919 − 0.298i)13-s + (0.309 − 0.951i)16-s + (0.171 − 0.236i)19-s − 1.79i·21-s + (−0.809 − 0.587i)25-s + (0.309 − 0.951i)27-s + (1.71 + 0.555i)28-s + (−0.0961 − 0.295i)31-s + (0.809 + 0.587i)36-s + (0.691 − 0.502i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.906 + 0.421i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.906 + 0.421i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(363\)    =    \(3 \cdot 11^{2}\)
Sign: $0.906 + 0.421i$
Analytic conductor: \(2.89856\)
Root analytic conductor: \(1.70251\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{363} (215, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 363,\ (\ :1/2),\ 0.906 + 0.421i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.38666 - 0.306882i\)
\(L(\frac12)\) \(\approx\) \(1.38666 - 0.306882i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.40 + 1.01i)T \)
11 \( 1 \)
good2 \( 1 + (-1.61 + 1.17i)T^{2} \)
5 \( 1 + (4.04 + 2.93i)T^{2} \)
7 \( 1 + (-2.79 - 3.85i)T + (-2.16 + 6.65i)T^{2} \)
13 \( 1 + (-3.31 + 1.07i)T + (10.5 - 7.64i)T^{2} \)
17 \( 1 + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (-0.749 + 1.03i)T + (-5.87 - 18.0i)T^{2} \)
23 \( 1 - 23T^{2} \)
29 \( 1 + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (0.535 + 1.64i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (-4.20 + 3.05i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (12.6 + 38.9i)T^{2} \)
43 \( 1 - 13.0iT - 43T^{2} \)
47 \( 1 + (-14.5 - 44.6i)T^{2} \)
53 \( 1 + (42.8 - 31.1i)T^{2} \)
59 \( 1 + (-18.2 + 56.1i)T^{2} \)
61 \( 1 + (11.1 + 3.62i)T + (49.3 + 35.8i)T^{2} \)
67 \( 1 - 12.1T + 67T^{2} \)
71 \( 1 + (57.4 + 41.7i)T^{2} \)
73 \( 1 + (6.34 + 8.73i)T + (-22.5 + 69.4i)T^{2} \)
79 \( 1 + (16.8 - 5.48i)T + (63.9 - 46.4i)T^{2} \)
83 \( 1 + (-67.1 - 48.7i)T^{2} \)
89 \( 1 - 89T^{2} \)
97 \( 1 + (1.54 + 4.75i)T + (-78.4 + 57.0i)T^{2} \)
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

−11.36288996408862022379793049151, −10.89059638425344034563987715479, −9.649254808323520682021950327854, −8.362208711335789061035894125226, −7.57599558503702620795807096703, −6.21190373530112422069952423992, −5.79994693510816812450162134746, −4.79502638872803308811494589587, −2.53106071850071691508810276613, −1.45566096592699597305641205409, 1.43788373786166793048893486939, 3.60605228023278873069756716637, 4.33430249154427719672961154721, 5.66921326197346016166245523918, 6.81004502290377389652117740828, 7.55542612424607705695469294535, 8.596294370714126511664645700881, 10.02810489719052653581250219477, 10.82254155673941430234715262069, 11.31644497112443510327365129891

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