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Properties

Label 2-819-91.74-c1-0-9
Degree $2$
Conductor $819$
Sign $-0.338 - 0.941i$
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  − 0.281·2-s − 1.92·4-s + (2.04 + 3.54i)5-s + (−1.79 − 1.94i)7-s + 1.10·8-s + (−0.575 − 0.996i)10-s + (−0.678 − 1.17i)11-s + (2.99 + 2.00i)13-s + (0.504 + 0.547i)14-s + 3.53·16-s + 6.60·17-s + (−1.75 + 3.04i)19-s + (−3.92 − 6.80i)20-s + (0.190 + 0.330i)22-s − 5.23·23-s + ⋯
L(s)  = 1  − 0.198·2-s − 0.960·4-s + (0.914 + 1.58i)5-s + (−0.677 − 0.735i)7-s + 0.389·8-s + (−0.182 − 0.315i)10-s + (−0.204 − 0.354i)11-s + (0.830 + 0.557i)13-s + (0.134 + 0.146i)14-s + 0.882·16-s + 1.60·17-s + (−0.402 + 0.698i)19-s + (−0.878 − 1.52i)20-s + (0.0406 + 0.0704i)22-s − 1.09·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.338 - 0.941i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.338 - 0.941i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $-0.338 - 0.941i$
Analytic conductor: \(6.53974\)
Root analytic conductor: \(2.55729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{819} (802, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 819,\ (\ :1/2),\ -0.338 - 0.941i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.574659 + 0.817257i\)
\(L(\frac12)\) \(\approx\) \(0.574659 + 0.817257i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.79 + 1.94i)T \)
13 \( 1 + (-2.99 - 2.00i)T \)
good2 \( 1 + 0.281T + 2T^{2} \)
5 \( 1 + (-2.04 - 3.54i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (0.678 + 1.17i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 - 6.60T + 17T^{2} \)
19 \( 1 + (1.75 - 3.04i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + 5.23T + 23T^{2} \)
29 \( 1 + (4.00 - 6.93i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (3.61 - 6.26i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 4.47T + 37T^{2} \)
41 \( 1 + (-0.766 + 1.32i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.847 - 1.46i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (2.69 + 4.67i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (0.981 - 1.69i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 - 2.10T + 59T^{2} \)
61 \( 1 + (1.94 - 3.37i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.76 + 4.78i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-5.21 - 9.02i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (1.90 - 3.29i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (2.59 + 4.49i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 12.2T + 83T^{2} \)
89 \( 1 - 16.8T + 89T^{2} \)
97 \( 1 + (0.760 + 1.31i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.42169587182257378504834415775, −9.827484660666867565188942317351, −9.004314803834448159838789007560, −7.87257463638196365056076214057, −7.02850543554977294272173201308, −6.14110252465974113116113865070, −5.42200841193119910719018672814, −3.70336837171711614949970056166, −3.35263373476329599160389228265, −1.59084004924050677637440850497, 0.56440730389144175122063956289, 1.95667250287271180096651978450, 3.63285665109503022637241382601, 4.72135919316144793061398717470, 5.65823971218078323524458035813, 5.96167774893362264801575965800, 7.86215894598281682675694901805, 8.395914731693673486595189471615, 9.400231702476024809579324792629, 9.542615621651385147339788537690

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