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Properties

Label 2-363-11.3-c1-0-1
Degree $2$
Conductor $363$
Sign $-0.469 - 0.882i$
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  + (0.640 − 0.465i)2-s + (0.309 + 0.951i)3-s + (−0.424 + 1.30i)4-s + (−2.72 − 1.98i)5-s + (0.640 + 0.465i)6-s + (−0.780 + 2.40i)7-s + (0.825 + 2.54i)8-s + (−0.809 + 0.587i)9-s − 2.67·10-s − 1.37·12-s + (−4.72 + 3.43i)13-s + (0.618 + 1.90i)14-s + (1.04 − 3.20i)15-s + (−0.507 − 0.368i)16-s + (2.16 + 1.57i)17-s + (−0.244 + 0.753i)18-s + ⋯
L(s)  = 1  + (0.453 − 0.329i)2-s + (0.178 + 0.549i)3-s + (−0.212 + 0.652i)4-s + (−1.22 − 0.886i)5-s + (0.261 + 0.190i)6-s + (−0.294 + 0.907i)7-s + (0.291 + 0.898i)8-s + (−0.269 + 0.195i)9-s − 0.844·10-s − 0.396·12-s + (−1.31 + 0.952i)13-s + (0.165 + 0.508i)14-s + (0.269 − 0.828i)15-s + (−0.126 − 0.0922i)16-s + (0.524 + 0.380i)17-s + (−0.0577 + 0.177i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.469 - 0.882i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.469 - 0.882i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(363\)    =    \(3 \cdot 11^{2}\)
Sign: $-0.469 - 0.882i$
Analytic conductor: \(2.89856\)
Root analytic conductor: \(1.70251\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{363} (124, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 363,\ (\ :1/2),\ -0.469 - 0.882i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.487257 + 0.811327i\)
\(L(\frac12)\) \(\approx\) \(0.487257 + 0.811327i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.309 - 0.951i)T \)
11 \( 1 \)
good2 \( 1 + (-0.640 + 0.465i)T + (0.618 - 1.90i)T^{2} \)
5 \( 1 + (2.72 + 1.98i)T + (1.54 + 4.75i)T^{2} \)
7 \( 1 + (0.780 - 2.40i)T + (-5.66 - 4.11i)T^{2} \)
13 \( 1 + (4.72 - 3.43i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (-2.16 - 1.57i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (-0.290 - 0.893i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 - 2T + 23T^{2} \)
29 \( 1 + (-0.244 + 0.753i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (1.31 - 0.956i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-1.54 + 4.75i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (-3.36 - 10.3i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 6.63T + 43T^{2} \)
47 \( 1 + (3.93 + 12.1i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (-3.33 + 2.41i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (1.85 - 5.70i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (-4.84 - 3.51i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + 1.11T + 67T^{2} \)
71 \( 1 + (-8.69 - 6.31i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (2.82 - 8.70i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (-3.32 + 2.41i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (1.52 + 1.10i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + 0.627T + 89T^{2} \)
97 \( 1 + (8.48 - 6.16i)T + (29.9 - 92.2i)T^{2} \)
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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

−11.97663768300446330777269044560, −11.21211595636438214027286984362, −9.700030715122904142628911612950, −8.887116730131215245537041386624, −8.185337135395066024267705511938, −7.25923895911084520143787519428, −5.42890037863529505318734439049, −4.54837480578173902220939230639, −3.78545087316832722549747445997, −2.53898598041433881755967970756, 0.54342189078466628748981980560, 2.93766257978848011739561374825, 4.02232239845444830468545885196, 5.20228016391967731989204452068, 6.53300201701359643232738048421, 7.36017471726892485910620323785, 7.75294038368010207311574088057, 9.408715212549200981601070532859, 10.36844518623831580701042809717, 11.03572820982431825611533999411

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