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Properties

Label 2-819-91.41-c1-0-17
Degree $2$
Conductor $819$
Sign $0.874 + 0.484i$
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  + (−1.96 − 0.525i)2-s + (1.84 + 1.06i)4-s + (1.56 + 1.56i)5-s + (2.60 + 0.462i)7-s + (−0.187 − 0.187i)8-s + (−2.25 − 3.90i)10-s + (1.16 − 4.33i)11-s + (3.51 − 0.786i)13-s + (−4.87 − 2.27i)14-s + (−1.86 − 3.22i)16-s + (1.31 − 2.27i)17-s + (−6.01 + 1.61i)19-s + (1.22 + 4.55i)20-s + (−4.56 + 7.90i)22-s + (4.58 − 2.64i)23-s + ⋯
L(s)  = 1  + (−1.38 − 0.371i)2-s + (0.922 + 0.532i)4-s + (0.700 + 0.700i)5-s + (0.984 + 0.174i)7-s + (−0.0661 − 0.0661i)8-s + (−0.712 − 1.23i)10-s + (0.350 − 1.30i)11-s + (0.975 − 0.218i)13-s + (−1.30 − 0.609i)14-s + (−0.465 − 0.805i)16-s + (0.318 − 0.552i)17-s + (−1.37 + 0.369i)19-s + (0.273 + 1.01i)20-s + (−0.972 + 1.68i)22-s + (0.956 − 0.552i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.874 + 0.484i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.874 + 0.484i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $0.874 + 0.484i$
Analytic conductor: \(6.53974\)
Root analytic conductor: \(2.55729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{819} (496, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 819,\ (\ :1/2),\ 0.874 + 0.484i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.992087 - 0.256174i\)
\(L(\frac12)\) \(\approx\) \(0.992087 - 0.256174i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.60 - 0.462i)T \)
13 \( 1 + (-3.51 + 0.786i)T \)
good2 \( 1 + (1.96 + 0.525i)T + (1.73 + i)T^{2} \)
5 \( 1 + (-1.56 - 1.56i)T + 5iT^{2} \)
11 \( 1 + (-1.16 + 4.33i)T + (-9.52 - 5.5i)T^{2} \)
17 \( 1 + (-1.31 + 2.27i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (6.01 - 1.61i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + (-4.58 + 2.64i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.06 - 3.57i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-2.44 - 2.44i)T + 31iT^{2} \)
37 \( 1 + (-0.0290 + 0.108i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + (-2.03 + 7.58i)T + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (5.47 + 3.16i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (5.82 - 5.82i)T - 47iT^{2} \)
53 \( 1 + 9.24T + 53T^{2} \)
59 \( 1 + (-1.27 - 4.77i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (-3.33 - 1.92i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-10.1 - 2.72i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + (3.56 + 13.3i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 + (-11.8 + 11.8i)T - 73iT^{2} \)
79 \( 1 + 13.3T + 79T^{2} \)
83 \( 1 + (-5.15 - 5.15i)T + 83iT^{2} \)
89 \( 1 + (-7.93 - 2.12i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (-1.93 + 0.518i)T + (84.0 - 48.5i)T^{2} \)
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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

−10.36584080301203021769114394753, −9.180417218960231423136763772332, −8.570865583548956053918128559693, −8.059706678126710824798293179269, −6.82014725397086080981188399802, −6.03685177644051923465385367900, −4.87718704177452819507758065633, −3.26971636122518853160183255412, −2.15214451186512516494184895007, −1.01009469147605144975675898873, 1.25399105895389018834935996625, 1.93417320415185393893975403398, 4.14654514078246604574527483379, 4.97495152368382755755498775286, 6.27034214976932875997542105800, 7.00508446160370117772097678330, 8.145377117501067113274631150515, 8.484882670162997509686645558283, 9.463530603023080345612487672371, 9.926677308347378890835645059831

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