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Properties

Label 2-363-33.17-c1-0-27
Degree $2$
Conductor $363$
Sign $0.399 - 0.916i$
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  + (−0.587 − 1.80i)2-s + (−0.945 − 1.45i)3-s + (−1.30 + 0.951i)4-s + (−2.48 − 0.809i)5-s + (−2.06 + 2.56i)6-s + (−0.427 − 0.587i)7-s + (−0.587 − 0.427i)8-s + (−1.21 + 2.74i)9-s + 4.97i·10-s + (2.61 + i)12-s + (2.92 − 0.951i)13-s + (−0.812 + 1.11i)14-s + (1.18 + 4.37i)15-s + (−1.42 + 4.39i)16-s + (0.812 − 2.5i)17-s + (5.67 + 0.577i)18-s + ⋯
L(s)  = 1  + (−0.415 − 1.27i)2-s + (−0.546 − 0.837i)3-s + (−0.654 + 0.475i)4-s + (−1.11 − 0.361i)5-s + (−0.844 + 1.04i)6-s + (−0.161 − 0.222i)7-s + (−0.207 − 0.150i)8-s + (−0.403 + 0.914i)9-s + 1.57i·10-s + (0.755 + 0.288i)12-s + (0.811 − 0.263i)13-s + (−0.217 + 0.298i)14-s + (0.304 + 1.13i)15-s + (−0.356 + 1.09i)16-s + (0.197 − 0.606i)17-s + (1.33 + 0.136i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.399 - 0.916i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.399 - 0.916i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(363\)    =    \(3 \cdot 11^{2}\)
Sign: $0.399 - 0.916i$
Analytic conductor: \(2.89856\)
Root analytic conductor: \(1.70251\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{363} (215, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 363,\ (\ :1/2),\ 0.399 - 0.916i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.193569 + 0.126844i\)
\(L(\frac12)\) \(\approx\) \(0.193569 + 0.126844i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.945 + 1.45i)T \)
11 \( 1 \)
good2 \( 1 + (0.587 + 1.80i)T + (-1.61 + 1.17i)T^{2} \)
5 \( 1 + (2.48 + 0.809i)T + (4.04 + 2.93i)T^{2} \)
7 \( 1 + (0.427 + 0.587i)T + (-2.16 + 6.65i)T^{2} \)
13 \( 1 + (-2.92 + 0.951i)T + (10.5 - 7.64i)T^{2} \)
17 \( 1 + (-0.812 + 2.5i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (2.5 - 3.44i)T + (-5.87 - 18.0i)T^{2} \)
23 \( 1 + 1.76iT - 23T^{2} \)
29 \( 1 + (3.07 - 2.23i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (0.263 + 0.812i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (2.42 - 1.76i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (2.48 + 1.80i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + 1.62iT - 43T^{2} \)
47 \( 1 + (4.30 - 5.92i)T + (-14.5 - 44.6i)T^{2} \)
53 \( 1 + (4.61 - 1.5i)T + (42.8 - 31.1i)T^{2} \)
59 \( 1 + (-1.53 - 2.11i)T + (-18.2 + 56.1i)T^{2} \)
61 \( 1 + (-4.04 - 1.31i)T + (49.3 + 35.8i)T^{2} \)
67 \( 1 + 8.32T + 67T^{2} \)
71 \( 1 + (9.82 + 3.19i)T + (57.4 + 41.7i)T^{2} \)
73 \( 1 + (8.94 + 12.3i)T + (-22.5 + 69.4i)T^{2} \)
79 \( 1 + (10.1 - 3.30i)T + (63.9 - 46.4i)T^{2} \)
83 \( 1 + (4.47 - 13.7i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 - 9.47iT - 89T^{2} \)
97 \( 1 + (2.04 + 6.29i)T + (-78.4 + 57.0i)T^{2} \)
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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

−10.93216635565718996029800628900, −10.12876839868603678489707953392, −8.784289723998152938733779396973, −8.058474989034690691340951257069, −6.98458987307398271759834036506, −5.82902941566661183632816569619, −4.28784193248554387997154953220, −3.14493119866640208778400768995, −1.54792979158296001989880057030, −0.19705575038837811389160996860, 3.31887690127537012266958993132, 4.40805636460600303528102390698, 5.66875815980641033234239967818, 6.48157938335551249148765252424, 7.40770613640362991902657910687, 8.452581497938795356557121271287, 9.093554105498877643416991799297, 10.27530706909340407419272419185, 11.34544225627189491891603152972, 11.74622434797966725468246611649

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