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Properties

Label 2-819-91.5-c1-0-13
Degree $2$
Conductor $819$
Sign $-0.185 - 0.982i$
Analytic cond. $6.53974$
Root an. cond. $2.55729$
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  + (0.211 + 0.788i)2-s + (1.15 − 0.666i)4-s + (−3.03 + 0.814i)5-s + (2.28 + 1.32i)7-s + (1.92 + 1.92i)8-s + (−1.28 − 2.22i)10-s + (0.131 − 0.491i)11-s + (−1.73 + 3.15i)13-s + (−0.562 + 2.08i)14-s + (0.221 − 0.382i)16-s + (0.606 + 1.05i)17-s + (−1.72 + 0.461i)19-s + (−2.96 + 2.96i)20-s + 0.415·22-s + (4.51 + 2.60i)23-s + ⋯
L(s)  = 1  + (0.149 + 0.557i)2-s + (0.577 − 0.333i)4-s + (−1.35 + 0.364i)5-s + (0.865 + 0.501i)7-s + (0.680 + 0.680i)8-s + (−0.406 − 0.703i)10-s + (0.0396 − 0.148i)11-s + (−0.481 + 0.876i)13-s + (−0.150 + 0.557i)14-s + (0.0552 − 0.0957i)16-s + (0.147 + 0.254i)17-s + (−0.394 + 0.105i)19-s + (−0.662 + 0.662i)20-s + 0.0885·22-s + (0.940 + 0.543i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.185 - 0.982i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.185 - 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $-0.185 - 0.982i$
Analytic conductor: \(6.53974\)
Root analytic conductor: \(2.55729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{819} (460, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 819,\ (\ :1/2),\ -0.185 - 0.982i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.02598 + 1.23806i\)
\(L(\frac12)\) \(\approx\) \(1.02598 + 1.23806i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 + (-2.28 - 1.32i)T \)
13 \( 1 + (1.73 - 3.15i)T \)
good2 \( 1 + (-0.211 - 0.788i)T + (-1.73 + i)T^{2} \)
5 \( 1 + (3.03 - 0.814i)T + (4.33 - 2.5i)T^{2} \)
11 \( 1 + (-0.131 + 0.491i)T + (-9.52 - 5.5i)T^{2} \)
17 \( 1 + (-0.606 - 1.05i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.72 - 0.461i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + (-4.51 - 2.60i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 1.64T + 29T^{2} \)
31 \( 1 + (0.976 - 3.64i)T + (-26.8 - 15.5i)T^{2} \)
37 \( 1 + (2.66 - 0.715i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (-5.55 - 5.55i)T + 41iT^{2} \)
43 \( 1 - 7.46iT - 43T^{2} \)
47 \( 1 + (-1.26 - 4.73i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (4.30 + 7.46i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-2.41 - 0.648i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (9.09 + 5.25i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-7.15 - 1.91i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + (0.840 - 0.840i)T - 71iT^{2} \)
73 \( 1 + (-2.36 - 0.632i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (-6.20 + 10.7i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-7.31 - 7.31i)T + 83iT^{2} \)
89 \( 1 + (2.52 + 9.42i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (2.93 + 2.93i)T + 97iT^{2} \)
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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

−10.90165261416551477631495494629, −9.549657542836903521623318320572, −8.451460923342750358253182689908, −7.76341562288327908371974886376, −7.12828125702692574854440391402, −6.24968143104812811878051130800, −5.11433656301470085160753104328, −4.35254377543811160981339413129, −3.01947320415557428571498834343, −1.66346974522274691053143859050, 0.78658499875331833403375818836, 2.36232600313705826275911546718, 3.59479617410183259281249261955, 4.30769243120290281850431902384, 5.25330965201237717500101023303, 6.86233464871144016923337235081, 7.60534131396551699777101943702, 8.017908466185195766886216696700, 9.075867039089558367069639470169, 10.49330469769917626195648157936

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