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Properties

Label 2-819-13.10-c1-0-23
Degree $2$
Conductor $819$
Sign $0.388 + 0.921i$
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  + (−1.10 + 0.638i)2-s + (−0.185 + 0.320i)4-s + 1.81i·5-s + (−0.866 − 0.5i)7-s − 3.02i·8-s + (−1.15 − 2.00i)10-s + (2.40 − 1.38i)11-s + (−3.58 − 0.402i)13-s + 1.27·14-s + (1.56 + 2.70i)16-s + (−1.37 + 2.37i)17-s + (−5.08 − 2.93i)19-s + (−0.582 − 0.336i)20-s + (−1.77 + 3.07i)22-s + (−3.49 − 6.06i)23-s + ⋯
L(s)  = 1  + (−0.781 + 0.451i)2-s + (−0.0925 + 0.160i)4-s + 0.811i·5-s + (−0.327 − 0.188i)7-s − 1.06i·8-s + (−0.366 − 0.634i)10-s + (0.725 − 0.418i)11-s + (−0.993 − 0.111i)13-s + 0.341·14-s + (0.390 + 0.675i)16-s + (−0.332 + 0.576i)17-s + (−1.16 − 0.673i)19-s + (−0.130 − 0.0751i)20-s + (−0.378 + 0.654i)22-s + (−0.729 − 1.26i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.388 + 0.921i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.388 + 0.921i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $0.388 + 0.921i$
Analytic conductor: \(6.53974\)
Root analytic conductor: \(2.55729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{819} (127, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 819,\ (\ :1/2),\ 0.388 + 0.921i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.314220 - 0.208525i\)
\(L(\frac12)\) \(\approx\) \(0.314220 - 0.208525i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.866 + 0.5i)T \)
13 \( 1 + (3.58 + 0.402i)T \)
good2 \( 1 + (1.10 - 0.638i)T + (1 - 1.73i)T^{2} \)
5 \( 1 - 1.81iT - 5T^{2} \)
11 \( 1 + (-2.40 + 1.38i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (1.37 - 2.37i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (5.08 + 2.93i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (3.49 + 6.06i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.75 + 3.04i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 2.06iT - 31T^{2} \)
37 \( 1 + (-1.50 + 0.871i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (5.51 - 3.18i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-4.55 + 7.88i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 6.65iT - 47T^{2} \)
53 \( 1 - 10.4T + 53T^{2} \)
59 \( 1 + (2.66 + 1.53i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (0.540 - 0.936i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-4.34 + 2.50i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (2.35 + 1.35i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + 7.67iT - 73T^{2} \)
79 \( 1 + 15.7T + 79T^{2} \)
83 \( 1 + 7.97iT - 83T^{2} \)
89 \( 1 + (-13.9 + 8.03i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (12.3 + 7.11i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.17733235528970283581711693660, −8.995850848705974414077495109033, −8.530075606143860606025555177557, −7.44067858649056804142389667947, −6.75455623631831514287777832077, −6.18040634955680006917956796492, −4.51165537962353871441141081935, −3.61738742485878534229315738229, −2.37882001188310837286177206633, −0.25205694616840489896030635245, 1.35572138111012357285759942144, 2.43904405147010210590610918500, 4.10585438386365942172850295240, 5.02093810068373621845072382180, 5.93660848783539032477143340130, 7.12307747416926555917973046687, 8.136045671583237863812824435157, 8.972168717081438902915745774313, 9.518436830376410772610965189660, 10.10681407156062606473534707279

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