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Properties

Label 2-363-33.26-c2-0-6
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: $\chi_{363} (323, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 363,\ (\ :1),\ -0.963 + 0.267i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.106717 - 0.784800i\)
\(L(\frac12)\) \(\approx\) \(0.106717 - 0.784800i\)
\(L(2)\) 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

−11.69271621721740349219118734138, −10.66004717207871506667743222408, −9.747758376688754424247469352694, −8.792224541799642404553931261366, −7.962086950949002927398560835688, −6.77981596370083332212026648687, −6.04820905403717942216354253376, −5.52943213395852447491210244221, −3.77510639394207807116291138444, −1.61669734055957316196072079550, 0.61601508348244066536339224098, 1.35154415023593399940493674181, 3.23932380770804799800872355489, 4.69449429595528740349512170040, 5.74144120695146771348793328778, 6.90685570578568787378512763840, 8.146235083294992347059942192431, 9.131216092893165170675911521150, 10.18422753914151638763547440483, 10.72077000140284573589517864417

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