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Properties

Label 2-363-3.2-c2-0-19
Degree $2$
Conductor $363$
Sign $-0.933 - 0.358i$
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  + 2.00i·2-s + (2.80 + 1.07i)3-s − 0.0168·4-s + 5.48i·5-s + (−2.15 + 5.61i)6-s − 5.59·7-s + 7.98i·8-s + (6.68 + 6.02i)9-s − 10.9·10-s + (−0.0470 − 0.0180i)12-s + 9.71·13-s − 11.2i·14-s + (−5.89 + 15.3i)15-s − 16.0·16-s − 17.8i·17-s + (−12.0 + 13.4i)18-s + ⋯
L(s)  = 1  + 1.00i·2-s + (0.933 + 0.358i)3-s − 0.00420·4-s + 1.09i·5-s + (−0.359 + 0.935i)6-s − 0.798·7-s + 0.997i·8-s + (0.742 + 0.669i)9-s − 1.09·10-s + (−0.00392 − 0.00150i)12-s + 0.747·13-s − 0.800i·14-s + (−0.393 + 1.02i)15-s − 1.00·16-s − 1.05i·17-s + (−0.670 + 0.744i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.933 - 0.358i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.933 - 0.358i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(363\)    =    \(3 \cdot 11^{2}\)
Sign: $-0.933 - 0.358i$
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: \((2,\ 363,\ (\ :1),\ -0.933 - 0.358i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.424724 + 2.29084i\)
\(L(\frac12)\) \(\approx\) \(0.424724 + 2.29084i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-2.80 - 1.07i)T \)
11 \( 1 \)
good2 \( 1 - 2.00iT - 4T^{2} \)
5 \( 1 - 5.48iT - 25T^{2} \)
7 \( 1 + 5.59T + 49T^{2} \)
13 \( 1 - 9.71T + 169T^{2} \)
17 \( 1 + 17.8iT - 289T^{2} \)
19 \( 1 + 18.6T + 361T^{2} \)
23 \( 1 + 12.3iT - 529T^{2} \)
29 \( 1 + 2.47iT - 841T^{2} \)
31 \( 1 - 49.2T + 961T^{2} \)
37 \( 1 + 39.3T + 1.36e3T^{2} \)
41 \( 1 - 56.5iT - 1.68e3T^{2} \)
43 \( 1 - 43.9T + 1.84e3T^{2} \)
47 \( 1 - 57.7iT - 2.20e3T^{2} \)
53 \( 1 + 43.1iT - 2.80e3T^{2} \)
59 \( 1 + 90.0iT - 3.48e3T^{2} \)
61 \( 1 - 30.8T + 3.72e3T^{2} \)
67 \( 1 - 34.0T + 4.48e3T^{2} \)
71 \( 1 + 37.5iT - 5.04e3T^{2} \)
73 \( 1 - 12.1T + 5.32e3T^{2} \)
79 \( 1 - 63.1T + 6.24e3T^{2} \)
83 \( 1 - 9.70iT - 6.88e3T^{2} \)
89 \( 1 + 34.1iT - 7.92e3T^{2} \)
97 \( 1 + 37.8T + 9.40e3T^{2} \)
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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

−11.35470408670810906085005826963, −10.58111952159743509709104494406, −9.667660899288366184783824699782, −8.613111753642202576082110198512, −7.79869915179920845117292433219, −6.74726345725707561490004292993, −6.33273226217817592568669712670, −4.73876299912131343109112824937, −3.28570437862469921442626770240, −2.44380054823068300488565697917, 0.971086856268067084644013677326, 2.12159354534412558259364250564, 3.43012148705407126269920530475, 4.23518596951904833312922315802, 6.06174753122701626167992155235, 7.03106626093104867117775325336, 8.404643960285319914443122106430, 8.941282369902924241476671782475, 9.940147172000030886841576167318, 10.68775233752954046922463205731

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