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Properties

Label 2-1078-77.54-c1-0-20
Degree $2$
Conductor $1078$
Sign $-0.339 - 0.940i$
Analytic cond. $8.60787$
Root an. cond. $2.93391$
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  + (0.866 + 0.5i)2-s + (−2.24 + 1.29i)3-s + (0.499 + 0.866i)4-s + (3.06 + 1.76i)5-s − 2.59·6-s + 0.999i·8-s + (1.86 − 3.23i)9-s + (1.76 + 3.06i)10-s + (3.22 + 0.789i)11-s + (−2.24 − 1.29i)12-s + 5.01·13-s − 9.18·15-s + (−0.5 + 0.866i)16-s + (1.94 + 3.37i)17-s + (3.23 − 1.86i)18-s + (2.32 − 4.03i)19-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (−1.29 + 0.749i)3-s + (0.249 + 0.433i)4-s + (1.37 + 0.791i)5-s − 1.05·6-s + 0.353i·8-s + (0.622 − 1.07i)9-s + (0.559 + 0.969i)10-s + (0.971 + 0.238i)11-s + (−0.648 − 0.374i)12-s + 1.39·13-s − 2.37·15-s + (−0.125 + 0.216i)16-s + (0.472 + 0.817i)17-s + (0.762 − 0.440i)18-s + (0.534 − 0.925i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.339 - 0.940i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.339 - 0.940i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1078\)    =    \(2 \cdot 7^{2} \cdot 11\)
Sign: $-0.339 - 0.940i$
Analytic conductor: \(8.60787\)
Root analytic conductor: \(2.93391\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1078} (901, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1078,\ (\ :1/2),\ -0.339 - 0.940i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.176970777\)
\(L(\frac12)\) \(\approx\) \(2.176970777\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 - 0.5i)T \)
7 \( 1 \)
11 \( 1 + (-3.22 - 0.789i)T \)
good3 \( 1 + (2.24 - 1.29i)T + (1.5 - 2.59i)T^{2} \)
5 \( 1 + (-3.06 - 1.76i)T + (2.5 + 4.33i)T^{2} \)
13 \( 1 - 5.01T + 13T^{2} \)
17 \( 1 + (-1.94 - 3.37i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.32 + 4.03i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.779 + 1.35i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 0.100iT - 29T^{2} \)
31 \( 1 + (-0.242 + 0.139i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.352 - 0.610i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 2.94T + 41T^{2} \)
43 \( 1 + 9.03iT - 43T^{2} \)
47 \( 1 + (5.68 + 3.28i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (2.77 + 4.79i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (12.5 - 7.26i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (5.56 - 9.63i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.99 + 3.45i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 8.45T + 71T^{2} \)
73 \( 1 + (1.94 + 3.37i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-6.66 - 3.84i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + 2.87T + 83T^{2} \)
89 \( 1 + (-5.15 - 2.97i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + 16.7iT - 97T^{2} \)
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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

−10.37042187361276518797792995723, −9.522292828117042182504179737337, −8.658368742646357740097891078848, −7.10599660898007263767252138075, −6.32690547054724075464264244258, −5.98300450056559571247615319615, −5.19255821623109414526763361172, −4.17176747686315466000404131997, −3.17164829036902421129227043456, −1.56795644452009039168990427575, 1.17934812278454468052750856203, 1.54685544601342892446318750917, 3.32367514428817037961589745801, 4.71439995216007052201296177600, 5.49457589836251584047098586397, 6.14665999959554569433617540267, 6.49828454300801946560812260744, 7.82100888188261501271630796116, 9.092946993440403418196398027833, 9.684736026414065668542657918639

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