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Properties

Label 2-363-33.14-c2-0-13
Degree $2$
Conductor $363$
Sign $-0.963 - 0.267i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.55 + 2.14i)2-s + (−2.99 − 0.0770i)3-s + (−0.927 + 2.85i)4-s + (1.66 − 2.28i)5-s + (−4.49 − 6.53i)6-s + (−2.31 + 7.11i)7-s + (2.51 − 0.817i)8-s + (8.98 + 0.461i)9-s + 7.48·10-s + (3.00 − 8.48i)12-s + (−18.1 + 13.1i)13-s + (−18.8 + 6.11i)14-s + (−5.16 + 6.73i)15-s + (15.3 + 11.1i)16-s + (−12.4 + 17.1i)17-s + (12.9 + 19.9i)18-s + ⋯
L(s)  = 1  + (0.777 + 1.07i)2-s + (−0.999 − 0.0256i)3-s + (−0.231 + 0.713i)4-s + (0.332 − 0.457i)5-s + (−0.749 − 1.08i)6-s + (−0.330 + 1.01i)7-s + (0.314 − 0.102i)8-s + (0.998 + 0.0513i)9-s + 0.748·10-s + (0.250 − 0.707i)12-s + (−1.39 + 1.01i)13-s + (−1.34 + 0.437i)14-s + (−0.344 + 0.448i)15-s + (0.960 + 0.697i)16-s + (−0.731 + 1.00i)17-s + (0.721 + 1.10i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.963 - 0.267i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.963 - 0.267i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(363\)    =    \(3 \cdot 11^{2}\)
Sign: $-0.963 - 0.267i$
Analytic conductor: \(9.89103\)
Root analytic conductor: \(3.14500\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{363} (245, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 363,\ (\ :1),\ -0.963 - 0.267i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.192295 + 1.41414i\)
\(L(\frac12)\) \(\approx\) \(0.192295 + 1.41414i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (2.99 + 0.0770i)T \)
11 \( 1 \)
good2 \( 1 + (-1.55 - 2.14i)T + (-1.23 + 3.80i)T^{2} \)
5 \( 1 + (-1.66 + 2.28i)T + (-7.72 - 23.7i)T^{2} \)
7 \( 1 + (2.31 - 7.11i)T + (-39.6 - 28.8i)T^{2} \)
13 \( 1 + (18.1 - 13.1i)T + (52.2 - 160. i)T^{2} \)
17 \( 1 + (12.4 - 17.1i)T + (-89.3 - 274. i)T^{2} \)
19 \( 1 + (-4.62 - 14.2i)T + (-292. + 212. i)T^{2} \)
23 \( 1 + 31.1iT - 529T^{2} \)
29 \( 1 + (20.1 + 6.54i)T + (680. + 494. i)T^{2} \)
31 \( 1 + (24.2 - 17.6i)T + (296. - 913. i)T^{2} \)
37 \( 1 + (3.09 - 9.51i)T + (-1.10e3 - 804. i)T^{2} \)
41 \( 1 + (-40.2 + 13.0i)T + (1.35e3 - 988. i)T^{2} \)
43 \( 1 + 14.9T + 1.84e3T^{2} \)
47 \( 1 + (34.9 - 11.3i)T + (1.78e3 - 1.29e3i)T^{2} \)
53 \( 1 + (-24.9 - 34.3i)T + (-868. + 2.67e3i)T^{2} \)
59 \( 1 + (-32.2 - 10.4i)T + (2.81e3 + 2.04e3i)T^{2} \)
61 \( 1 + (-78.7 - 57.1i)T + (1.14e3 + 3.53e3i)T^{2} \)
67 \( 1 + 42T + 4.48e3T^{2} \)
71 \( 1 + (-38.2 + 52.6i)T + (-1.55e3 - 4.79e3i)T^{2} \)
73 \( 1 + (-23.1 + 71.1i)T + (-4.31e3 - 3.13e3i)T^{2} \)
79 \( 1 + (18.1 - 13.1i)T + (1.92e3 - 5.93e3i)T^{2} \)
83 \( 1 + (12.4 - 17.1i)T + (-2.12e3 - 6.55e3i)T^{2} \)
89 \( 1 + 62.2iT - 7.92e3T^{2} \)
97 \( 1 + (59.8 - 43.4i)T + (2.90e3 - 8.94e3i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.00790681804449702473687714173, −10.77757922469706711661269777589, −9.765967483730335217593369118596, −8.783808212465352098184226889804, −7.41365011004344371323485628445, −6.55240975773034096589493287594, −5.80661917144166662114800429022, −5.05124258610240554730888099220, −4.17558721127699387802214861616, −1.91595766202122502797321334661, 0.55102439787766550832426963870, 2.33408060228472348439913508196, 3.58837754878095002964026633772, 4.74077413919793464057790638492, 5.49380533180678434698851278424, 7.00062761540253127268757526925, 7.48651511136994822260527134003, 9.720959526242715956161386104240, 10.08760768616262581696675041659, 11.13448318909764567116688187570

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