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Properties

Label 2-363-3.2-c2-0-31
Degree $2$
Conductor $363$
Sign $-0.914 + 0.404i$
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  − 3.58i·2-s + (−2.74 + 1.21i)3-s − 8.86·4-s + 2.08i·5-s + (4.35 + 9.84i)6-s + 8.86·7-s + 17.4i·8-s + (6.05 − 6.66i)9-s + 7.48·10-s + (24.3 − 10.7i)12-s − 1.64·13-s − 31.7i·14-s + (−2.53 − 5.72i)15-s + 27.1·16-s − 12.4i·17-s + (−23.8 − 21.7i)18-s + ⋯
L(s)  = 1  − 1.79i·2-s + (−0.914 + 0.404i)3-s − 2.21·4-s + 0.417i·5-s + (0.725 + 1.64i)6-s + 1.26·7-s + 2.18i·8-s + (0.672 − 0.740i)9-s + 0.748·10-s + (2.02 − 0.897i)12-s − 0.126·13-s − 2.27i·14-s + (−0.168 − 0.381i)15-s + 1.69·16-s − 0.730i·17-s + (−1.32 − 1.20i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.914 + 0.404i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.914 + 0.404i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(363\)    =    \(3 \cdot 11^{2}\)
Sign: $-0.914 + 0.404i$
Analytic conductor: \(9.89103\)
Root analytic conductor: \(3.14500\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{363} (122, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 363,\ (\ :1),\ -0.914 + 0.404i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.225373 - 1.06600i\)
\(L(\frac12)\) \(\approx\) \(0.225373 - 1.06600i\)
\(L(2)\) 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 \( 1 + 3.58iT - 4T^{2} \)
5 \( 1 - 2.08iT - 25T^{2} \)
7 \( 1 - 8.86T + 49T^{2} \)
13 \( 1 + 1.64T + 169T^{2} \)
17 \( 1 + 12.4iT - 289T^{2} \)
19 \( 1 - 10.5T + 361T^{2} \)
23 \( 1 + 20.3iT - 529T^{2} \)
29 \( 1 + 11.6iT - 841T^{2} \)
31 \( 1 + 23.3T + 961T^{2} \)
37 \( 1 - 7.24T + 1.36e3T^{2} \)
41 \( 1 + 38.8iT - 1.68e3T^{2} \)
43 \( 1 + 15.8T + 1.84e3T^{2} \)
47 \( 1 + 45.3iT - 2.20e3T^{2} \)
53 \( 1 + 40.3iT - 2.80e3T^{2} \)
59 \( 1 + 113. iT - 3.48e3T^{2} \)
61 \( 1 - 77.5T + 3.72e3T^{2} \)
67 \( 1 - 62.9T + 4.48e3T^{2} \)
71 \( 1 - 10.2iT - 5.04e3T^{2} \)
73 \( 1 + 74.6T + 5.32e3T^{2} \)
79 \( 1 - 86.3T + 6.24e3T^{2} \)
83 \( 1 + 11.8iT - 6.88e3T^{2} \)
89 \( 1 + 74.5iT - 7.92e3T^{2} \)
97 \( 1 + 77.1T + 9.40e3T^{2} \)
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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

−11.01956492622740552485701428529, −10.28007189142008394448855299394, −9.457913076536808325812927953877, −8.398217604691159545146284814054, −6.97714297413743609865948763613, −5.30134406254928547130330363829, −4.66656792438108857843492540725, −3.52782114546494455855929767098, −2.08549562349983896742448902270, −0.66715295302583485843994878347, 1.27433738315227101277751307437, 4.32647844953056788902623661266, 5.14893809324030290490712389730, 5.76647736863471304746961160916, 6.89521707176952013460181585549, 7.69336437359550980683723798352, 8.353866258757996187652439946099, 9.419469814690992898183890055363, 10.72751265528556524829775686008, 11.67941578606627281266212797923

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