Properties

Label 2-819-117.2-c1-0-28
Degree $2$
Conductor $819$
Sign $0.996 - 0.0850i$
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  + (−0.996 − 0.996i)2-s + (0.260 + 1.71i)3-s − 0.0156i·4-s + (−0.114 + 0.428i)5-s + (1.44 − 1.96i)6-s + (0.258 − 0.965i)7-s + (−2.00 + 2.00i)8-s + (−2.86 + 0.892i)9-s + (0.540 − 0.312i)10-s + (3.75 − 3.75i)11-s + (0.0267 − 0.00406i)12-s + (2.50 + 2.59i)13-s + (−1.21 + 0.704i)14-s + (−0.763 − 0.0848i)15-s + 3.96·16-s + (−3.82 + 6.63i)17-s + ⋯
L(s)  = 1  + (−0.704 − 0.704i)2-s + (0.150 + 0.988i)3-s − 0.00780i·4-s + (−0.0513 + 0.191i)5-s + (0.590 − 0.802i)6-s + (0.0978 − 0.365i)7-s + (−0.709 + 0.709i)8-s + (−0.954 + 0.297i)9-s + (0.171 − 0.0987i)10-s + (1.13 − 1.13i)11-s + (0.00771 − 0.00117i)12-s + (0.695 + 0.718i)13-s + (−0.326 + 0.188i)14-s + (−0.197 − 0.0218i)15-s + 0.992·16-s + (−0.928 + 1.60i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 - 0.0850i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.996 - 0.0850i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $0.996 - 0.0850i$
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: \((2,\ 819,\ (\ :1/2),\ 0.996 - 0.0850i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.09214 + 0.0465390i\)
\(L(\frac12)\) \(\approx\) \(1.09214 + 0.0465390i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.260 - 1.71i)T \)
7 \( 1 + (-0.258 + 0.965i)T \)
13 \( 1 + (-2.50 - 2.59i)T \)
good2 \( 1 + (0.996 + 0.996i)T + 2iT^{2} \)
5 \( 1 + (0.114 - 0.428i)T + (-4.33 - 2.5i)T^{2} \)
11 \( 1 + (-3.75 + 3.75i)T - 11iT^{2} \)
17 \( 1 + (3.82 - 6.63i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.960 + 3.58i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (-1.99 + 3.45i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 1.11iT - 29T^{2} \)
31 \( 1 + (-6.63 - 1.77i)T + (26.8 + 15.5i)T^{2} \)
37 \( 1 + (-0.352 + 1.31i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + (-0.870 + 0.233i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (-5.44 + 3.14i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-2.60 - 9.70i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 - 11.0iT - 53T^{2} \)
59 \( 1 + (3.64 - 3.64i)T - 59iT^{2} \)
61 \( 1 + (-3.55 - 6.16i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.434 + 1.62i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (-9.15 + 2.45i)T + (61.4 - 35.5i)T^{2} \)
73 \( 1 + (-10.5 - 10.5i)T + 73iT^{2} \)
79 \( 1 + (-7.57 + 13.1i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-9.35 + 2.50i)T + (71.8 - 41.5i)T^{2} \)
89 \( 1 + (1.88 + 0.505i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (2.54 + 0.682i)T + (84.0 + 48.5i)T^{2} \)
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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

−10.52421186442115372403114423816, −9.211557102190767098681994941622, −8.932792489881561523021543994969, −8.294799339698855085038894211913, −6.56466284093708951119467410179, −5.98051204731470278583694375872, −4.56815627450512491242384077205, −3.73930053274386540868251726967, −2.62540533410045662621504512147, −1.10824689954194233009910386385, 0.865224981221439156321374686813, 2.39444066213435859396416093616, 3.67337294391953495655075702023, 5.10809576838832723194750040492, 6.42407835155194976449940236185, 6.77523812419421165298914533299, 7.72685820975676081744593807955, 8.380143846843033645850781416900, 9.151905056036673258590072825300, 9.747334405253756425484277249484

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