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Properties

Label 2-819-117.2-c1-0-32
Degree $2$
Conductor $819$
Sign $0.166 + 0.986i$
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  + (−0.993 − 0.993i)2-s + (−0.927 + 1.46i)3-s − 0.0271i·4-s + (−0.251 + 0.940i)5-s + (2.37 − 0.532i)6-s + (−0.258 + 0.965i)7-s + (−2.01 + 2.01i)8-s + (−1.28 − 2.71i)9-s + (1.18 − 0.683i)10-s + (−1.51 + 1.51i)11-s + (0.0396 + 0.0251i)12-s + (−2.35 − 2.72i)13-s + (1.21 − 0.702i)14-s + (−1.14 − 1.24i)15-s + 3.94·16-s + (−0.0983 + 0.170i)17-s + ⋯
L(s)  = 1  + (−0.702 − 0.702i)2-s + (−0.535 + 0.844i)3-s − 0.0135i·4-s + (−0.112 + 0.420i)5-s + (0.969 − 0.217i)6-s + (−0.0978 + 0.365i)7-s + (−0.711 + 0.711i)8-s + (−0.427 − 0.904i)9-s + (0.374 − 0.216i)10-s + (−0.457 + 0.457i)11-s + (0.0114 + 0.00725i)12-s + (−0.653 − 0.757i)13-s + (0.325 − 0.187i)14-s + (−0.294 − 0.320i)15-s + 0.986·16-s + (−0.0238 + 0.0413i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.166 + 0.986i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.166 + 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $0.166 + 0.986i$
Analytic conductor: \(6.53974\)
Root analytic conductor: \(2.55729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{819} (470, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 819,\ (\ :1/2),\ 0.166 + 0.986i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.408643 - 0.345398i\)
\(L(\frac12)\) \(\approx\) \(0.408643 - 0.345398i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.927 - 1.46i)T \)
7 \( 1 + (0.258 - 0.965i)T \)
13 \( 1 + (2.35 + 2.72i)T \)
good2 \( 1 + (0.993 + 0.993i)T + 2iT^{2} \)
5 \( 1 + (0.251 - 0.940i)T + (-4.33 - 2.5i)T^{2} \)
11 \( 1 + (1.51 - 1.51i)T - 11iT^{2} \)
17 \( 1 + (0.0983 - 0.170i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.646 + 2.41i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (-3.20 + 5.54i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 5.84iT - 29T^{2} \)
31 \( 1 + (3.24 + 0.869i)T + (26.8 + 15.5i)T^{2} \)
37 \( 1 + (-1.53 + 5.73i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + (-0.670 + 0.179i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (-10.8 + 6.28i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-1.67 - 6.24i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + 6.91iT - 53T^{2} \)
59 \( 1 + (-2.97 + 2.97i)T - 59iT^{2} \)
61 \( 1 + (3.34 + 5.78i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.12 + 11.6i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (-5.26 + 1.41i)T + (61.4 - 35.5i)T^{2} \)
73 \( 1 + (6.18 + 6.18i)T + 73iT^{2} \)
79 \( 1 + (-2.60 + 4.50i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-11.7 + 3.14i)T + (71.8 - 41.5i)T^{2} \)
89 \( 1 + (11.7 + 3.15i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (-17.5 - 4.71i)T + (84.0 + 48.5i)T^{2} \)
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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

−10.28745234491211491727342612539, −9.266164257716813462984659145844, −8.901454646068321822405633759514, −7.59217291243844067332212384022, −6.48560258006120292364128701697, −5.44139443414776814011853870278, −4.78771850999289621272894929386, −3.25909617692193119768795519803, −2.39196755652009861830969129720, −0.44607510838300300093861584963, 1.00999434052179504377876972214, 2.71437344265480798105962714297, 4.21017916670341624437231303526, 5.44646965352443906801800655872, 6.31121984579815666855332341966, 7.19553483945709793910704807597, 7.70719547436780157991369789074, 8.521219739088522277702093672077, 9.340443544419557809860482339901, 10.30449045979604760202970334157

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