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Properties

Label 2-819-91.16-c1-0-42
Degree $2$
Conductor $819$
Sign $0.137 + 0.990i$
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  + 1.85·2-s + 1.45·4-s + (1.41 − 2.45i)5-s + (−2.59 − 0.514i)7-s − 1.01·8-s + (2.63 − 4.56i)10-s + (2.68 − 4.64i)11-s + (−1.80 − 3.11i)13-s + (−4.82 − 0.955i)14-s − 4.79·16-s − 0.835·17-s + (1.94 + 3.36i)19-s + (2.05 − 3.56i)20-s + (4.98 − 8.62i)22-s + 4.10·23-s + ⋯
L(s)  = 1  + 1.31·2-s + 0.726·4-s + (0.633 − 1.09i)5-s + (−0.980 − 0.194i)7-s − 0.359·8-s + (0.832 − 1.44i)10-s + (0.808 − 1.40i)11-s + (−0.501 − 0.865i)13-s + (−1.28 − 0.255i)14-s − 1.19·16-s − 0.202·17-s + (0.445 + 0.771i)19-s + (0.460 − 0.797i)20-s + (1.06 − 1.83i)22-s + 0.856·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.137 + 0.990i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.137 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $0.137 + 0.990i$
Analytic conductor: \(6.53974\)
Root analytic conductor: \(2.55729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{819} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 819,\ (\ :1/2),\ 0.137 + 0.990i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.06817 - 1.80148i\)
\(L(\frac12)\) \(\approx\) \(2.06817 - 1.80148i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.59 + 0.514i)T \)
13 \( 1 + (1.80 + 3.11i)T \)
good2 \( 1 - 1.85T + 2T^{2} \)
5 \( 1 + (-1.41 + 2.45i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-2.68 + 4.64i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + 0.835T + 17T^{2} \)
19 \( 1 + (-1.94 - 3.36i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 4.10T + 23T^{2} \)
29 \( 1 + (-2.20 - 3.82i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (0.678 + 1.17i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 9.07T + 37T^{2} \)
41 \( 1 + (1.58 + 2.75i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-1.24 + 2.16i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-0.0166 + 0.0288i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-5.18 - 8.97i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 - 1.74T + 59T^{2} \)
61 \( 1 + (3.04 + 5.26i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.41 - 2.44i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (7.26 - 12.5i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (2.16 + 3.74i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (2.96 - 5.13i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 3.75T + 83T^{2} \)
89 \( 1 - 12.3T + 89T^{2} \)
97 \( 1 + (-8.56 + 14.8i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.976116145996928414950155984506, −9.144341317778045421628300878286, −8.585868120211073446538899725823, −7.15483411676003205327393924229, −5.97439077725329640671624100300, −5.72335761336631657554840391274, −4.68500898652609831430988960432, −3.63656859905629865265520937322, −2.84239614340077620739795179023, −0.904015663620155247033369539779, 2.26785237566122549848172805727, 2.98438888477594891623488164886, 4.13286806033564515533411581830, 4.94886783740188969560996452465, 6.20530171573989189130127364498, 6.65565179619101266653183814798, 7.26371896226612121499438992376, 9.203637443946128706654707253087, 9.506719677362625151721374607269, 10.44141067326292987062525021389

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