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Properties

Label 2-819-117.2-c1-0-34
Degree $2$
Conductor $819$
Sign $-0.601 - 0.798i$
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  + (1.70 + 1.70i)2-s + (1.58 − 0.697i)3-s + 3.84i·4-s + (−0.668 + 2.49i)5-s + (3.90 + 1.51i)6-s + (0.258 − 0.965i)7-s + (−3.15 + 3.15i)8-s + (2.02 − 2.21i)9-s + (−5.40 + 3.12i)10-s + (−3.92 + 3.92i)11-s + (2.68 + 6.09i)12-s + (−2.50 + 2.59i)13-s + (2.09 − 1.20i)14-s + (0.679 + 4.42i)15-s − 3.09·16-s + (2.36 − 4.08i)17-s + ⋯
L(s)  = 1  + (1.20 + 1.20i)2-s + (0.915 − 0.402i)3-s + 1.92i·4-s + (−0.298 + 1.11i)5-s + (1.59 + 0.620i)6-s + (0.0978 − 0.365i)7-s + (−1.11 + 1.11i)8-s + (0.676 − 0.736i)9-s + (−1.71 + 0.987i)10-s + (−1.18 + 1.18i)11-s + (0.773 + 1.76i)12-s + (−0.694 + 0.719i)13-s + (0.559 − 0.323i)14-s + (0.175 + 1.14i)15-s − 0.774·16-s + (0.572 − 0.991i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.601 - 0.798i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.601 - 0.798i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $-0.601 - 0.798i$
Analytic conductor: \(6.53974\)
Root analytic conductor: \(2.55729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{819} (470, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 819,\ (\ :1/2),\ -0.601 - 0.798i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.59943 + 3.20838i\)
\(L(\frac12)\) \(\approx\) \(1.59943 + 3.20838i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.58 + 0.697i)T \)
7 \( 1 + (-0.258 + 0.965i)T \)
13 \( 1 + (2.50 - 2.59i)T \)
good2 \( 1 + (-1.70 - 1.70i)T + 2iT^{2} \)
5 \( 1 + (0.668 - 2.49i)T + (-4.33 - 2.5i)T^{2} \)
11 \( 1 + (3.92 - 3.92i)T - 11iT^{2} \)
17 \( 1 + (-2.36 + 4.08i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.461 - 1.72i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (-3.54 + 6.13i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 2.71iT - 29T^{2} \)
31 \( 1 + (-7.98 - 2.13i)T + (26.8 + 15.5i)T^{2} \)
37 \( 1 + (0.852 - 3.17i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + (-8.21 + 2.20i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (8.80 - 5.08i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (2.94 + 11.0i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + 0.0631iT - 53T^{2} \)
59 \( 1 + (-5.08 + 5.08i)T - 59iT^{2} \)
61 \( 1 + (4.38 + 7.59i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.439 + 1.64i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (-0.771 + 0.206i)T + (61.4 - 35.5i)T^{2} \)
73 \( 1 + (3.25 + 3.25i)T + 73iT^{2} \)
79 \( 1 + (6.50 - 11.2i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-4.57 + 1.22i)T + (71.8 - 41.5i)T^{2} \)
89 \( 1 + (-4.86 - 1.30i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (15.0 + 4.02i)T + (84.0 + 48.5i)T^{2} \)
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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

−10.36347785841807426936800856534, −9.695811536724023766583757855741, −8.214893839217266533490109521598, −7.66290962125316283181359116193, −6.89915471664863657152581208694, −6.67452579682991928103755663438, −5.04962957945833114035931156236, −4.38811140829610015966299137223, −3.18347609356327979408192181832, −2.47929547023021047230872390053, 1.20505459415458888372437841205, 2.67579258373465094844128928984, 3.26190265806481707964494682583, 4.37308886979969780696790784603, 5.16523291184017890601143695575, 5.70953484205534673707862053739, 7.72805119964518872618366731364, 8.339803359011956832958135817166, 9.251622763630501603715519213077, 10.18816627898734379668130814963

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