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Properties

Label 2-363-33.20-c2-0-62
Degree $2$
Conductor $363$
Sign $-0.944 - 0.329i$
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  + (2.42 − 1.76i)3-s + (−3.23 − 2.35i)4-s + (−8.89 − 6.46i)7-s + (2.78 − 8.55i)9-s − 12·12-s + (−6.79 + 20.9i)13-s + (4.94 + 15.2i)16-s + (−8.89 + 6.46i)19-s − 33·21-s + (−20.2 + 14.6i)25-s + (−8.34 − 25.6i)27-s + (13.5 + 41.8i)28-s + (18.2 − 56.1i)31-s + (−29.1 + 21.1i)36-s + (−38.0 − 27.6i)37-s + ⋯
L(s)  = 1  + (0.809 − 0.587i)3-s + (−0.809 − 0.587i)4-s + (−1.27 − 0.923i)7-s + (0.309 − 0.951i)9-s − 12-s + (−0.522 + 1.60i)13-s + (0.309 + 0.951i)16-s + (−0.468 + 0.340i)19-s − 1.57·21-s + (−0.809 + 0.587i)25-s + (−0.309 − 0.951i)27-s + (0.485 + 1.49i)28-s + (0.588 − 1.81i)31-s + (−0.809 + 0.587i)36-s + (−1.02 − 0.746i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.944 - 0.329i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.944 - 0.329i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(363\)    =    \(3 \cdot 11^{2}\)
Sign: $-0.944 - 0.329i$
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: \((2,\ 363,\ (\ :1),\ -0.944 - 0.329i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.0886125 + 0.522254i\)
\(L(\frac12)\) \(\approx\) \(0.0886125 + 0.522254i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-2.42 + 1.76i)T \)
11 \( 1 \)
good2 \( 1 + (3.23 + 2.35i)T^{2} \)
5 \( 1 + (20.2 - 14.6i)T^{2} \)
7 \( 1 + (8.89 + 6.46i)T + (15.1 + 46.6i)T^{2} \)
13 \( 1 + (6.79 - 20.9i)T + (-136. - 99.3i)T^{2} \)
17 \( 1 + (233. - 169. i)T^{2} \)
19 \( 1 + (8.89 - 6.46i)T + (111. - 343. i)T^{2} \)
23 \( 1 - 529T^{2} \)
29 \( 1 + (-259. - 799. i)T^{2} \)
31 \( 1 + (-18.2 + 56.1i)T + (-777. - 564. i)T^{2} \)
37 \( 1 + (38.0 + 27.6i)T + (423. + 1.30e3i)T^{2} \)
41 \( 1 + (-519. + 1.59e3i)T^{2} \)
43 \( 1 + 22T + 1.84e3T^{2} \)
47 \( 1 + (-682. + 2.10e3i)T^{2} \)
53 \( 1 + (2.27e3 + 1.65e3i)T^{2} \)
59 \( 1 + (-1.07e3 - 3.31e3i)T^{2} \)
61 \( 1 + (37.3 + 115. i)T + (-3.01e3 + 2.18e3i)T^{2} \)
67 \( 1 + 13T + 4.48e3T^{2} \)
71 \( 1 + (4.07e3 - 2.96e3i)T^{2} \)
73 \( 1 + (115. + 84.0i)T + (1.64e3 + 5.06e3i)T^{2} \)
79 \( 1 + (-3.39 + 10.4i)T + (-5.04e3 - 3.66e3i)T^{2} \)
83 \( 1 + (5.57e3 - 4.04e3i)T^{2} \)
89 \( 1 - 7.92e3T^{2} \)
97 \( 1 + (52.2 - 160. i)T + (-7.61e3 - 5.53e3i)T^{2} \)
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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

−10.36827546584540522539331625290, −9.532552216235379203994909387275, −9.157125606275151791180121297068, −7.85792312951267434527733885020, −6.85884717993630688574348426436, −6.10315086149856215439040895351, −4.36386590187500226160834867377, −3.60025121847501490875417958842, −1.87851899334028750759060389365, −0.21102579953631851036730017923, 2.77684223518322526236873720519, 3.33817518468887647130614571136, 4.68394630644303628108994932619, 5.70902638178937529884098021660, 7.21011017170529806795696564489, 8.394459455719004668100875834226, 8.800452573673245776875218290366, 9.904314683628729662158076934525, 10.27831023163684159125780519372, 12.02475977227485026020919240417

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