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Properties

Label 2-1026-9.7-c1-0-12
Degree $2$
Conductor $1026$
Sign $-0.256 + 0.966i$
Analytic cond. $8.19265$
Root an. cond. $2.86228$
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  + (0.5 − 0.866i)2-s + (−0.499 − 0.866i)4-s + (0.573 + 0.992i)5-s + (0.817 − 1.41i)7-s − 0.999·8-s + 1.14·10-s + (1.10 − 1.91i)11-s + (−2.18 − 3.78i)13-s + (−0.817 − 1.41i)14-s + (−0.5 + 0.866i)16-s + 1.42·17-s − 19-s + (0.573 − 0.992i)20-s + (−1.10 − 1.91i)22-s + (2.16 + 3.75i)23-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.249 − 0.433i)4-s + (0.256 + 0.443i)5-s + (0.308 − 0.535i)7-s − 0.353·8-s + 0.362·10-s + (0.333 − 0.576i)11-s + (−0.605 − 1.04i)13-s + (−0.218 − 0.378i)14-s + (−0.125 + 0.216i)16-s + 0.345·17-s − 0.229·19-s + (0.128 − 0.221i)20-s + (−0.235 − 0.407i)22-s + (0.452 + 0.783i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.256 + 0.966i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.256 + 0.966i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1026\)    =    \(2 \cdot 3^{3} \cdot 19\)
Sign: $-0.256 + 0.966i$
Analytic conductor: \(8.19265\)
Root analytic conductor: \(2.86228\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1026} (343, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1026,\ (\ :1/2),\ -0.256 + 0.966i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.870185437\)
\(L(\frac12)\) \(\approx\) \(1.870185437\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 + 0.866i)T \)
3 \( 1 \)
19 \( 1 + T \)
good5 \( 1 + (-0.573 - 0.992i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (-0.817 + 1.41i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-1.10 + 1.91i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (2.18 + 3.78i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 - 1.42T + 17T^{2} \)
23 \( 1 + (-2.16 - 3.75i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.700 + 1.21i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (3.66 + 6.34i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 6.94T + 37T^{2} \)
41 \( 1 + (2.64 + 4.57i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-4.37 + 7.58i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-1.53 + 2.66i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 5.16T + 53T^{2} \)
59 \( 1 + (-0.653 - 1.13i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.535 + 0.927i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.57 + 6.18i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 12.0T + 71T^{2} \)
73 \( 1 - 6.78T + 73T^{2} \)
79 \( 1 + (5.40 - 9.36i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (2.59 - 4.50i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 7.62T + 89T^{2} \)
97 \( 1 + (7.77 - 13.4i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.920692268417660358911086414603, −9.074982503792396384121621003656, −7.986192897243339135928751398352, −7.22586648995472516557585646615, −6.08580193555730850797916439851, −5.35854487605205465926108881387, −4.24682911315558510286421662160, −3.30251849364989862378766143861, −2.30934507305500006739678781385, −0.794415817568573804083545098189, 1.62568551335744669254642332632, 2.93011562917905091528471244162, 4.39537727237921970336180618768, 4.89878763835452345167117152919, 5.90395487196747909198379295600, 6.80606296689340428697591649684, 7.53168042609021233771653775708, 8.635604570842534962818326759054, 9.143767460623761653189541142625, 9.942508746298512927050504653099

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