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Properties

Label 2-819-117.49-c1-0-14
Degree $2$
Conductor $819$
Sign $0.893 - 0.448i$
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  + (−1.98 − 1.14i)2-s + (−1.71 + 0.259i)3-s + (1.62 + 2.81i)4-s + (1.57 + 0.911i)5-s + (3.69 + 1.44i)6-s + i·7-s − 2.86i·8-s + (2.86 − 0.888i)9-s + (−2.08 − 3.61i)10-s + (5.48 + 3.16i)11-s + (−3.51 − 4.39i)12-s + (−3.12 − 1.79i)13-s + (1.14 − 1.98i)14-s + (−2.94 − 1.15i)15-s + (−0.0323 + 0.0560i)16-s + (0.400 − 0.694i)17-s + ⋯
L(s)  = 1  + (−1.40 − 0.810i)2-s + (−0.988 + 0.149i)3-s + (0.812 + 1.40i)4-s + (0.706 + 0.407i)5-s + (1.50 + 0.590i)6-s + 0.377i·7-s − 1.01i·8-s + (0.955 − 0.296i)9-s + (−0.660 − 1.14i)10-s + (1.65 + 0.953i)11-s + (−1.01 − 1.26i)12-s + (−0.867 − 0.497i)13-s + (0.306 − 0.530i)14-s + (−0.759 − 0.297i)15-s + (−0.00809 + 0.0140i)16-s + (0.0972 − 0.168i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.893 - 0.448i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.893 - 0.448i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $0.893 - 0.448i$
Analytic conductor: \(6.53974\)
Root analytic conductor: \(2.55729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{819} (400, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 819,\ (\ :1/2),\ 0.893 - 0.448i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.619871 + 0.146771i\)
\(L(\frac12)\) \(\approx\) \(0.619871 + 0.146771i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.71 - 0.259i)T \)
7 \( 1 - iT \)
13 \( 1 + (3.12 + 1.79i)T \)
good2 \( 1 + (1.98 + 1.14i)T + (1 + 1.73i)T^{2} \)
5 \( 1 + (-1.57 - 0.911i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (-5.48 - 3.16i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (-0.400 + 0.694i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-6.09 - 3.52i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 - 1.19T + 23T^{2} \)
29 \( 1 + (2.42 - 4.19i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (7.58 + 4.37i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-2.63 + 1.52i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 - 12.7iT - 41T^{2} \)
43 \( 1 - 1.74T + 43T^{2} \)
47 \( 1 + (-1.65 + 0.956i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 - 6.89T + 53T^{2} \)
59 \( 1 + (-6.40 + 3.69i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + 1.86T + 61T^{2} \)
67 \( 1 + 4.93iT - 67T^{2} \)
71 \( 1 + (-0.0906 - 0.0523i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 - 1.03iT - 73T^{2} \)
79 \( 1 + (2.62 + 4.55i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-10.8 + 6.27i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (3.60 - 2.08i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 10.2iT - 97T^{2} \)
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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

−10.08184055707309753886470537427, −9.595972097255051473566670456609, −9.227900253697155523232622330912, −7.70978129503244464780579856204, −7.06782194856674519967409512184, −6.04276742520729201254223852271, −5.06036141780937468047677167239, −3.60927405436804423380226812215, −2.17442967020439478182448062319, −1.18803418065448372213334409027, 0.70360614838155385107779201855, 1.64493847486476143571659132993, 3.94632931086306509981955889288, 5.38227954533996684415730881191, 5.94759578578354961215229990319, 7.09000017956231252473239700878, 7.20231533447064178540326891060, 8.675796346661378799837029886107, 9.383095739802034372430414371906, 9.782448639235061949314582243340

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