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Properties

Label 2-819-117.103-c1-0-18
Degree $2$
Conductor $819$
Sign $-0.966 - 0.257i$
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  + (−1.34 + 0.777i)2-s + (−1.61 + 0.633i)3-s + (0.208 − 0.360i)4-s + (3.42 + 1.97i)5-s + (1.67 − 2.10i)6-s + (0.866 − 0.5i)7-s − 2.46i·8-s + (2.19 − 2.04i)9-s − 6.15·10-s + (−3.70 + 2.13i)11-s + (−0.106 + 0.712i)12-s + (−2.21 + 2.84i)13-s + (−0.777 + 1.34i)14-s + (−6.78 − 1.01i)15-s + (2.32 + 4.03i)16-s + 4.55·17-s + ⋯
L(s)  = 1  + (−0.951 + 0.549i)2-s + (−0.930 + 0.365i)3-s + (0.104 − 0.180i)4-s + (1.53 + 0.885i)5-s + (0.684 − 0.859i)6-s + (0.327 − 0.188i)7-s − 0.870i·8-s + (0.732 − 0.681i)9-s − 1.94·10-s + (−1.11 + 0.644i)11-s + (−0.0308 + 0.205i)12-s + (−0.613 + 0.789i)13-s + (−0.207 + 0.359i)14-s + (−1.75 − 0.262i)15-s + (0.582 + 1.00i)16-s + 1.10·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.966 - 0.257i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.966 - 0.257i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $-0.966 - 0.257i$
Analytic conductor: \(6.53974\)
Root analytic conductor: \(2.55729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{819} (337, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 819,\ (\ :1/2),\ -0.966 - 0.257i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0916190 + 0.699028i\)
\(L(\frac12)\) \(\approx\) \(0.0916190 + 0.699028i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.61 - 0.633i)T \)
7 \( 1 + (-0.866 + 0.5i)T \)
13 \( 1 + (2.21 - 2.84i)T \)
good2 \( 1 + (1.34 - 0.777i)T + (1 - 1.73i)T^{2} \)
5 \( 1 + (-3.42 - 1.97i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (3.70 - 2.13i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 - 4.55T + 17T^{2} \)
19 \( 1 - 0.0791iT - 19T^{2} \)
23 \( 1 + (-2.23 + 3.86i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3.77 - 6.54i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (2.43 + 1.40i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 - 10.8iT - 37T^{2} \)
41 \( 1 + (6.97 + 4.02i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2.72 - 4.72i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (0.831 - 0.480i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + 7.49T + 53T^{2} \)
59 \( 1 + (4.09 + 2.36i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.10 - 1.90i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.46 - 2.00i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 13.6iT - 71T^{2} \)
73 \( 1 - 7.88iT - 73T^{2} \)
79 \( 1 + (-2.97 - 5.16i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (2.51 - 1.45i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + 1.00iT - 89T^{2} \)
97 \( 1 + (12.7 - 7.37i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.29486243783320376761876990100, −9.891559878477423631746821429230, −9.238333359303246543031029851210, −7.976275569168197311331512924126, −6.93885060581022695403474574677, −6.62314518356510813699560370363, −5.47745421394565964308745886420, −4.70276279453606111666686345275, −3.03905163829799868613226104682, −1.52632121524955935237376944452, 0.56896287636476288714559732634, 1.61698211641038276357285273084, 2.61825962691709361394622871223, 5.03720100330460190737616746956, 5.39863607713289176671253711223, 5.98020594399095949980376297243, 7.58502860489302358001228711069, 8.252624288422469550743099530432, 9.260080272701638812301066404538, 10.01756505987556692860642764655

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