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Properties

Label 2-819-117.103-c1-0-24
Degree $2$
Conductor $819$
Sign $0.872 + 0.487i$
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  + (−2.16 + 1.25i)2-s + (−0.959 − 1.44i)3-s + (2.13 − 3.69i)4-s + (−2.59 − 1.49i)5-s + (3.88 + 1.92i)6-s + (−0.866 + 0.5i)7-s + 5.65i·8-s + (−1.15 + 2.76i)9-s + 7.48·10-s + (0.0836 − 0.0483i)11-s + (−7.36 + 0.466i)12-s + (3.50 − 0.844i)13-s + (1.25 − 2.16i)14-s + (0.327 + 5.17i)15-s + (−2.81 − 4.87i)16-s + 0.817·17-s + ⋯
L(s)  = 1  + (−1.53 + 0.884i)2-s + (−0.553 − 0.832i)3-s + (1.06 − 1.84i)4-s + (−1.15 − 0.668i)5-s + (1.58 + 0.785i)6-s + (−0.327 + 0.188i)7-s + 2.00i·8-s + (−0.386 + 0.922i)9-s + 2.36·10-s + (0.0252 − 0.0145i)11-s + (−2.12 + 0.134i)12-s + (0.972 − 0.234i)13-s + (0.334 − 0.579i)14-s + (0.0846 + 1.33i)15-s + (−0.704 − 1.21i)16-s + 0.198·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.872 + 0.487i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.872 + 0.487i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $0.872 + 0.487i$
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: \((2,\ 819,\ (\ :1/2),\ 0.872 + 0.487i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.343534 - 0.0894633i\)
\(L(\frac12)\) \(\approx\) \(0.343534 - 0.0894633i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.959 + 1.44i)T \)
7 \( 1 + (0.866 - 0.5i)T \)
13 \( 1 + (-3.50 + 0.844i)T \)
good2 \( 1 + (2.16 - 1.25i)T + (1 - 1.73i)T^{2} \)
5 \( 1 + (2.59 + 1.49i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (-0.0836 + 0.0483i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 - 0.817T + 17T^{2} \)
19 \( 1 - 4.88iT - 19T^{2} \)
23 \( 1 + (3.04 - 5.26i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.48 - 2.57i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-3.57 - 2.06i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + 7.18iT - 37T^{2} \)
41 \( 1 + (7.99 + 4.61i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (2.08 + 3.61i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-10.0 + 5.78i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + 1.69T + 53T^{2} \)
59 \( 1 + (-12.4 - 7.18i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.11 - 3.66i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (7.80 + 4.50i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 10.7iT - 71T^{2} \)
73 \( 1 + 2.77iT - 73T^{2} \)
79 \( 1 + (-2.25 - 3.90i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (4.93 - 2.85i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + 3.62iT - 89T^{2} \)
97 \( 1 + (-0.357 + 0.206i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.12197980200496136571684392059, −8.835385084716309166716900176774, −8.408732265908113771265425069379, −7.67412272793320487123249871377, −7.06087523706273121816900560597, −6.02895146313691490665186320595, −5.40445889598004154809468880866, −3.74965269752196326894969961479, −1.67043749477584510106043167858, −0.53890739674532420738306420480, 0.73349918507776660339895531547, 2.76650765887450948178164206002, 3.59296520115342712639990490707, 4.50292734061637558671342284021, 6.31347954098648795062462440051, 7.03646987279034126458439989645, 8.165519584992280881476569697678, 8.688894213833255559740856049592, 9.736932923390083163677441445777, 10.29776850869850234458512277825

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