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Properties

Label 2-363-33.5-c2-0-6
Degree $2$
Conductor $363$
Sign $-0.289 - 0.957i$
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  + (−2.91 + 0.945i)2-s + (−1.86 + 2.35i)3-s + (4.34 − 3.15i)4-s + (−6.31 − 2.05i)5-s + (3.20 − 8.60i)6-s + (−2.47 + 1.80i)7-s + (−2.45 + 3.37i)8-s + (−2.05 − 8.76i)9-s + 20.3·10-s + (−0.678 + 16.0i)12-s + (−5.01 − 15.4i)13-s + (5.50 − 7.58i)14-s + (16.5 − 11.0i)15-s + (−2.68 + 8.25i)16-s + (−0.766 − 0.248i)17-s + (14.2 + 23.5i)18-s + ⋯
L(s)  = 1  + (−1.45 + 0.472i)2-s + (−0.621 + 0.783i)3-s + (1.08 − 0.788i)4-s + (−1.26 − 0.410i)5-s + (0.533 − 1.43i)6-s + (−0.353 + 0.257i)7-s + (−0.306 + 0.422i)8-s + (−0.227 − 0.973i)9-s + 2.03·10-s + (−0.0565 + 1.34i)12-s + (−0.386 − 1.18i)13-s + (0.393 − 0.541i)14-s + (1.10 − 0.734i)15-s + (−0.167 + 0.515i)16-s + (−0.0450 − 0.0146i)17-s + (0.791 + 1.30i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.289 - 0.957i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.289 - 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(363\)    =    \(3 \cdot 11^{2}\)
Sign: $-0.289 - 0.957i$
Analytic conductor: \(9.89103\)
Root analytic conductor: \(3.14500\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{363} (269, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 363,\ (\ :1),\ -0.289 - 0.957i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.141358 + 0.190492i\)
\(L(\frac12)\) \(\approx\) \(0.141358 + 0.190492i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.86 - 2.35i)T \)
11 \( 1 \)
good2 \( 1 + (2.91 - 0.945i)T + (3.23 - 2.35i)T^{2} \)
5 \( 1 + (6.31 + 2.05i)T + (20.2 + 14.6i)T^{2} \)
7 \( 1 + (2.47 - 1.80i)T + (15.1 - 46.6i)T^{2} \)
13 \( 1 + (5.01 + 15.4i)T + (-136. + 99.3i)T^{2} \)
17 \( 1 + (0.766 + 0.248i)T + (233. + 169. i)T^{2} \)
19 \( 1 + (-16.7 - 12.1i)T + (111. + 343. i)T^{2} \)
23 \( 1 + 27.3iT - 529T^{2} \)
29 \( 1 + (-2.22 - 3.06i)T + (-259. + 799. i)T^{2} \)
31 \( 1 + (6.42 + 19.7i)T + (-777. + 564. i)T^{2} \)
37 \( 1 + (31.1 - 22.6i)T + (423. - 1.30e3i)T^{2} \)
41 \( 1 + (7.86 - 10.8i)T + (-519. - 1.59e3i)T^{2} \)
43 \( 1 + 43.4T + 1.84e3T^{2} \)
47 \( 1 + (11.6 - 16.0i)T + (-682. - 2.10e3i)T^{2} \)
53 \( 1 + (16.8 - 5.46i)T + (2.27e3 - 1.65e3i)T^{2} \)
59 \( 1 + (-25.5 - 35.2i)T + (-1.07e3 + 3.31e3i)T^{2} \)
61 \( 1 + (3.29 - 10.1i)T + (-3.01e3 - 2.18e3i)T^{2} \)
67 \( 1 - 72.2T + 4.48e3T^{2} \)
71 \( 1 + (2.44 + 0.794i)T + (4.07e3 + 2.96e3i)T^{2} \)
73 \( 1 + (36.7 - 26.7i)T + (1.64e3 - 5.06e3i)T^{2} \)
79 \( 1 + (-30.3 - 93.4i)T + (-5.04e3 + 3.66e3i)T^{2} \)
83 \( 1 + (30.3 + 9.85i)T + (5.57e3 + 4.04e3i)T^{2} \)
89 \( 1 + 18.5iT - 7.92e3T^{2} \)
97 \( 1 + (19.5 + 60.2i)T + (-7.61e3 + 5.53e3i)T^{2} \)
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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

−11.21508558499232023563126032308, −10.28226138470813272646742891615, −9.694697385307547395955578346610, −8.613175194769667503990150780875, −8.010748621106464782931098309604, −7.00359300822249542154290019926, −5.83861981224693910535067834420, −4.61541794357099156840574924629, −3.34808936904038608657254737007, −0.69796084898117227907664335517, 0.32670212836539773344081153887, 1.82784581193390095900187224554, 3.39018554338538434701286469353, 5.02631255312929856697680011580, 6.83894100788310750958164783796, 7.23933623614244302706727666181, 8.061208210912831117194042187212, 9.091252424327353827834635129789, 10.09208435851895357464768598768, 11.08965938365957286851750033990

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