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Properties

Label 2-363-33.8-c1-0-27
Degree $2$
Conductor $363$
Sign $-0.836 + 0.547i$
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  + (1.98 − 1.43i)2-s + (−1.28 − 1.15i)3-s + (1.23 − 3.80i)4-s + (1.01 − 1.40i)5-s + (−4.21 − 0.443i)6-s + (−1.34 − 0.437i)7-s + (−1.51 − 4.65i)8-s + (0.313 + 2.98i)9-s − 4.24i·10-s + (−6 + 3.46i)12-s + (1.66 + 2.28i)13-s + (−3.29 + 1.07i)14-s + (−2.93 + 0.623i)15-s + (−3.23 − 2.35i)16-s + (1.98 + 1.43i)17-s + (4.91 + 5.46i)18-s + ⋯
L(s)  = 1  + (1.40 − 1.01i)2-s + (−0.743 − 0.669i)3-s + (0.618 − 1.90i)4-s + (0.455 − 0.626i)5-s + (−1.72 − 0.181i)6-s + (−0.508 − 0.165i)7-s + (−0.535 − 1.64i)8-s + (0.104 + 0.994i)9-s − 1.34i·10-s + (−1.73 + 1.00i)12-s + (0.461 + 0.634i)13-s + (−0.880 + 0.286i)14-s + (−0.757 + 0.161i)15-s + (−0.809 − 0.587i)16-s + (0.480 + 0.349i)17-s + (1.15 + 1.28i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.836 + 0.547i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.836 + 0.547i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(363\)    =    \(3 \cdot 11^{2}\)
Sign: $-0.836 + 0.547i$
Analytic conductor: \(2.89856\)
Root analytic conductor: \(1.70251\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{363} (239, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 363,\ (\ :1/2),\ -0.836 + 0.547i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.649633 - 2.17806i\)
\(L(\frac12)\) \(\approx\) \(0.649633 - 2.17806i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.28 + 1.15i)T \)
11 \( 1 \)
good2 \( 1 + (-1.98 + 1.43i)T + (0.618 - 1.90i)T^{2} \)
5 \( 1 + (-1.01 + 1.40i)T + (-1.54 - 4.75i)T^{2} \)
7 \( 1 + (1.34 + 0.437i)T + (5.66 + 4.11i)T^{2} \)
13 \( 1 + (-1.66 - 2.28i)T + (-4.01 + 12.3i)T^{2} \)
17 \( 1 + (-1.98 - 1.43i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (2.68 - 0.874i)T + (15.3 - 11.1i)T^{2} \)
23 \( 1 + 8.66iT - 23T^{2} \)
29 \( 1 + (-1.51 + 4.65i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (2.42 - 1.76i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-0.927 + 2.85i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (-3.02 - 9.31i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 9.89iT - 43T^{2} \)
47 \( 1 + (-6.58 + 2.14i)T + (38.0 - 27.6i)T^{2} \)
53 \( 1 + (2.03 + 2.80i)T + (-16.3 + 50.4i)T^{2} \)
59 \( 1 + (-1.64 - 0.535i)T + (47.7 + 34.6i)T^{2} \)
61 \( 1 + (1.66 - 2.28i)T + (-18.8 - 58.0i)T^{2} \)
67 \( 1 + 3T + 67T^{2} \)
71 \( 1 + (-1.01 + 1.40i)T + (-21.9 - 67.5i)T^{2} \)
73 \( 1 + (-5.37 - 1.74i)T + (59.0 + 42.9i)T^{2} \)
79 \( 1 + (-5.81 - 8.00i)T + (-24.4 + 75.1i)T^{2} \)
83 \( 1 + (1.98 + 1.43i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 - 8.66iT - 89T^{2} \)
97 \( 1 + (10.5 - 7.64i)T + (29.9 - 92.2i)T^{2} \)
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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

−11.27358533828770833078911963868, −10.60408215382634333998133302689, −9.642968714459818190705847785464, −8.229227439766439356716371347028, −6.57980970397616512099513286825, −6.03439661416632535019843795623, −4.97861359226051589863810062732, −4.08373803683919368704197767990, −2.49496391190534923625688527556, −1.24987125262178150640442787349, 3.06712029934779376114327501788, 3.92976788271143378497829132019, 5.24756348464626653258148297858, 5.83237638522337031921458120367, 6.62667517242236531677196531719, 7.51920048621100745448146346433, 9.030384031412831952637044729342, 10.15038048695774940956278695879, 11.01193863237904527624404608838, 12.08061296124856556783767818211

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