↑

Properties

Label 2-363-33.26-c2-0-25
Degree $2$
Conductor $363$
Sign $0.992 + 0.120i$
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  + (−0.587 + 0.809i)2-s + (−2.74 + 1.21i)3-s + (0.927 + 2.85i)4-s + (−2.57 − 3.54i)5-s + (0.633 − 2.93i)6-s + (0.663 + 2.04i)7-s + (−6.65 − 2.16i)8-s + (6.06 − 6.64i)9-s + 4.38·10-s + (−6 − 6.70i)12-s + (−7.09 − 5.15i)13-s + (−2.04 − 0.663i)14-s + (11.3 + 6.61i)15-s + (−4.04 + 2.93i)16-s + (−3.52 − 4.85i)17-s + (1.81 + 8.81i)18-s + ⋯
L(s)  = 1  + (−0.293 + 0.404i)2-s + (−0.914 + 0.403i)3-s + (0.231 + 0.713i)4-s + (−0.515 − 0.709i)5-s + (0.105 − 0.488i)6-s + (0.0947 + 0.291i)7-s + (−0.832 − 0.270i)8-s + (0.674 − 0.738i)9-s + 0.438·10-s + (−0.5 − 0.559i)12-s + (−0.545 − 0.396i)13-s + (−0.145 − 0.0473i)14-s + (0.757 + 0.440i)15-s + (−0.252 + 0.183i)16-s + (−0.207 − 0.285i)17-s + (0.100 + 0.489i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 + 0.120i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.992 + 0.120i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(363\)    =    \(3 \cdot 11^{2}\)
Sign: $0.992 + 0.120i$
Analytic conductor: \(9.89103\)
Root analytic conductor: \(3.14500\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{363} (323, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 363,\ (\ :1),\ 0.992 + 0.120i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.762786 - 0.0462933i\)
\(L(\frac12)\) \(\approx\) \(0.762786 - 0.0462933i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (2.74 - 1.21i)T \)
11 \( 1 \)
good2 \( 1 + (0.587 - 0.809i)T + (-1.23 - 3.80i)T^{2} \)
5 \( 1 + (2.57 + 3.54i)T + (-7.72 + 23.7i)T^{2} \)
7 \( 1 + (-0.663 - 2.04i)T + (-39.6 + 28.8i)T^{2} \)
13 \( 1 + (7.09 + 5.15i)T + (52.2 + 160. i)T^{2} \)
17 \( 1 + (3.52 + 4.85i)T + (-89.3 + 274. i)T^{2} \)
19 \( 1 + (-9.61 + 29.6i)T + (-292. - 212. i)T^{2} \)
23 \( 1 - 26.4iT - 529T^{2} \)
29 \( 1 + (-45.0 + 14.6i)T + (680. - 494. i)T^{2} \)
31 \( 1 + (-24.9 - 18.1i)T + (296. + 913. i)T^{2} \)
37 \( 1 + (-9.14 - 28.1i)T + (-1.10e3 + 804. i)T^{2} \)
41 \( 1 + (41.4 + 13.4i)T + (1.35e3 + 988. i)T^{2} \)
43 \( 1 - 39.7T + 1.84e3T^{2} \)
47 \( 1 + (-4.08 - 1.32i)T + (1.78e3 + 1.29e3i)T^{2} \)
53 \( 1 + (-6.62 + 9.11i)T + (-868. - 2.67e3i)T^{2} \)
59 \( 1 + (-31.4 + 10.2i)T + (2.81e3 - 2.04e3i)T^{2} \)
61 \( 1 + (-22.1 + 16.0i)T + (1.14e3 - 3.53e3i)T^{2} \)
67 \( 1 - 70.7T + 4.48e3T^{2} \)
71 \( 1 + (51.8 + 71.3i)T + (-1.55e3 + 4.79e3i)T^{2} \)
73 \( 1 + (19.1 + 58.9i)T + (-4.31e3 + 3.13e3i)T^{2} \)
79 \( 1 + (51.0 + 37.0i)T + (1.92e3 + 5.93e3i)T^{2} \)
83 \( 1 + (64.7 + 89.1i)T + (-2.12e3 + 6.55e3i)T^{2} \)
89 \( 1 + 9.66iT - 7.92e3T^{2} \)
97 \( 1 + (81.0 + 58.8i)T + (2.90e3 + 8.94e3i)T^{2} \)
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

−11.58329353263971333940317854126, −10.23304712126726663397191321607, −9.216183469438467590519163211743, −8.414105843269839710488811353980, −7.35898508307546061199937267023, −6.52435483006978820555853805769, −5.22224861808084969906674447505, −4.40900630676348529849091071345, −2.97365789127574369987377098707, −0.52814406131388608565893398473, 1.06940221582915140371937246178, 2.52252849915060457801140801364, 4.25701555030026145937230963476, 5.52748657368504051953113187836, 6.49484489173905527434974613071, 7.21712551037800515944698707566, 8.368739570560797967726461490701, 9.899221312975930146556030316587, 10.41882077779593847594331277011, 11.17611549281848074177866604648

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