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Properties

Label 2-363-33.2-c1-0-24
Degree $2$
Conductor $363$
Sign $-0.455 + 0.890i$
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  + (0.465 − 1.43i)2-s + (0.479 − 1.66i)3-s + (−0.216 − 0.157i)4-s + (2.76 − 0.899i)5-s + (−2.16 − 1.46i)6-s + (−1.66 + 2.28i)7-s + (2.11 − 1.53i)8-s + (−2.54 − 1.59i)9-s − 4.38i·10-s + (−0.366 + 0.285i)12-s + (−0.852 − 0.277i)13-s + (2.50 + 3.44i)14-s + (−0.170 − 5.03i)15-s + (−1.37 − 4.24i)16-s + (0.806 + 2.48i)17-s + (−3.46 + 2.89i)18-s + ⋯
L(s)  = 1  + (0.329 − 1.01i)2-s + (0.276 − 0.960i)3-s + (−0.108 − 0.0787i)4-s + (1.23 − 0.402i)5-s + (−0.882 − 0.596i)6-s + (−0.628 + 0.864i)7-s + (0.746 − 0.542i)8-s + (−0.846 − 0.531i)9-s − 1.38i·10-s + (−0.105 + 0.0823i)12-s + (−0.236 − 0.0768i)13-s + (0.669 + 0.920i)14-s + (−0.0440 − 1.30i)15-s + (−0.344 − 1.06i)16-s + (0.195 + 0.601i)17-s + (−0.817 + 0.682i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.455 + 0.890i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.455 + 0.890i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(363\)    =    \(3 \cdot 11^{2}\)
Sign: $-0.455 + 0.890i$
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: \((2,\ 363,\ (\ :1/2),\ -0.455 + 0.890i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.09059 - 1.78331i\)
\(L(\frac12)\) \(\approx\) \(1.09059 - 1.78331i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.479 + 1.66i)T \)
11 \( 1 \)
good2 \( 1 + (-0.465 + 1.43i)T + (-1.61 - 1.17i)T^{2} \)
5 \( 1 + (-2.76 + 0.899i)T + (4.04 - 2.93i)T^{2} \)
7 \( 1 + (1.66 - 2.28i)T + (-2.16 - 6.65i)T^{2} \)
13 \( 1 + (0.852 + 0.277i)T + (10.5 + 7.64i)T^{2} \)
17 \( 1 + (-0.806 - 2.48i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (-1.43 - 1.98i)T + (-5.87 + 18.0i)T^{2} \)
23 \( 1 - 23T^{2} \)
29 \( 1 + (6.98 + 5.07i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (3.15 - 9.69i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (-2.86 - 2.07i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (3.65 - 2.65i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + 4.24iT - 43T^{2} \)
47 \( 1 + (1.25 + 1.72i)T + (-14.5 + 44.6i)T^{2} \)
53 \( 1 + (-6.27 - 2.03i)T + (42.8 + 31.1i)T^{2} \)
59 \( 1 + (-0.335 + 0.461i)T + (-18.2 - 56.1i)T^{2} \)
61 \( 1 + (-9.67 + 3.14i)T + (49.3 - 35.8i)T^{2} \)
67 \( 1 - 2T + 67T^{2} \)
71 \( 1 + (9.04 - 2.93i)T + (57.4 - 41.7i)T^{2} \)
73 \( 1 + (-1.88 + 2.59i)T + (-22.5 - 69.4i)T^{2} \)
79 \( 1 + (-6.98 - 2.27i)T + (63.9 + 46.4i)T^{2} \)
83 \( 1 + (-1.95 - 6.00i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + 12.4iT - 89T^{2} \)
97 \( 1 + (-1.77 + 5.45i)T + (-78.4 - 57.0i)T^{2} \)
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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

−11.42239532559256359604565723412, −10.17928566583562406934697566693, −9.466270875680072232810337553821, −8.525625695779350471415242670399, −7.25987847268898259513526041720, −6.17640751872664370630417148432, −5.36656387859892292424089598189, −3.49061486615824710858327994452, −2.40605402657046743797457768080, −1.56387259376780399414910856604, 2.36158475856762299068401307729, 3.82629591353241616320851104777, 5.12108716109552577623744973951, 5.85477470107295213628935818368, 6.86366867179076230183552681653, 7.69842168180416232748980051328, 9.178174523840569579133775424586, 9.826779621153939209114226398235, 10.58583994846208472901143156351, 11.38786252921533168002639937748

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