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Properties

Label 2-363-33.2-c1-0-12
Degree $2$
Conductor $363$
Sign $-0.0107 - 0.999i$
Analytic cond. $2.89856$
Root an. cond. $1.70251$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.756 + 2.32i)2-s + (1.72 − 0.181i)3-s + (−3.23 − 2.35i)4-s + (1.64 − 0.535i)5-s + (−0.882 + 4.14i)6-s + (0.831 − 1.14i)7-s + (3.96 − 2.87i)8-s + (2.93 − 0.623i)9-s + 4.24i·10-s + (−5.99 − 3.46i)12-s + (2.68 + 0.874i)13-s + (2.03 + 2.80i)14-s + (2.74 − 1.22i)15-s + (1.23 + 3.80i)16-s + (−0.756 − 2.32i)17-s + (−0.768 + 7.30i)18-s + ⋯
L(s)  = 1  + (−0.535 + 1.64i)2-s + (0.994 − 0.104i)3-s + (−1.61 − 1.17i)4-s + (0.736 − 0.239i)5-s + (−0.360 + 1.69i)6-s + (0.314 − 0.432i)7-s + (1.40 − 1.01i)8-s + (0.978 − 0.207i)9-s + 1.34i·10-s + (−1.73 − 0.999i)12-s + (0.746 + 0.242i)13-s + (0.544 + 0.749i)14-s + (0.707 − 0.315i)15-s + (0.309 + 0.951i)16-s + (−0.183 − 0.565i)17-s + (−0.181 + 1.72i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0107 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0107 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(363\)    =    \(3 \cdot 11^{2}\)
Sign: $-0.0107 - 0.999i$
Analytic conductor: \(2.89856\)
Root analytic conductor: \(1.70251\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{363} (233, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 363,\ (\ :1/2),\ -0.0107 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.08343 + 1.09510i\)
\(L(\frac12)\) \(\approx\) \(1.08343 + 1.09510i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.72 + 0.181i)T \)
11 \( 1 \)
good2 \( 1 + (0.756 - 2.32i)T + (-1.61 - 1.17i)T^{2} \)
5 \( 1 + (-1.64 + 0.535i)T + (4.04 - 2.93i)T^{2} \)
7 \( 1 + (-0.831 + 1.14i)T + (-2.16 - 6.65i)T^{2} \)
13 \( 1 + (-2.68 - 0.874i)T + (10.5 + 7.64i)T^{2} \)
17 \( 1 + (0.756 + 2.32i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (-1.66 - 2.28i)T + (-5.87 + 18.0i)T^{2} \)
23 \( 1 - 8.66iT - 23T^{2} \)
29 \( 1 + (3.96 + 2.87i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (-0.927 + 2.85i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (2.42 + 1.76i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (7.92 - 5.75i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + 9.89iT - 43T^{2} \)
47 \( 1 + (4.07 + 5.60i)T + (-14.5 + 44.6i)T^{2} \)
53 \( 1 + (3.29 + 1.07i)T + (42.8 + 31.1i)T^{2} \)
59 \( 1 + (1.01 - 1.40i)T + (-18.2 - 56.1i)T^{2} \)
61 \( 1 + (2.68 - 0.874i)T + (49.3 - 35.8i)T^{2} \)
67 \( 1 + 3T + 67T^{2} \)
71 \( 1 + (-1.64 + 0.535i)T + (57.4 - 41.7i)T^{2} \)
73 \( 1 + (3.32 - 4.57i)T + (-22.5 - 69.4i)T^{2} \)
79 \( 1 + (-9.41 - 3.05i)T + (63.9 + 46.4i)T^{2} \)
83 \( 1 + (-0.756 - 2.32i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + 8.66iT - 89T^{2} \)
97 \( 1 + (-4.01 + 12.3i)T + (-78.4 - 57.0i)T^{2} \)
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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

−11.60371072136778390833784984522, −10.07479805562057875170602534332, −9.456085258503206763727048781825, −8.748408602233513506580019972669, −7.79621652212894932288758628874, −7.20512354566858505760205324428, −6.08251212636879559330599865371, −5.12165354108957865085454669328, −3.73287183823944192570495316835, −1.59183072414271261169198607762, 1.59024812407170653310188896023, 2.54325550139373592872696154525, 3.51949486220212929934712672826, 4.76791885420543427168333781876, 6.47133560585424762041381707198, 8.089380730780450488804142166552, 8.740853496521558968933487476605, 9.453161220945038702119019727425, 10.37071506529304752443179617366, 10.86583333533002000736312114971

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