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Properties

Label 2-1078-77.54-c1-0-28
Degree $2$
Conductor $1078$
Sign $0.526 - 0.850i$
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 + (0.662 − 0.382i)3-s + (0.499 + 0.866i)4-s + (3.37 + 1.94i)5-s + 0.765·6-s + 0.999i·8-s + (−1.20 + 2.09i)9-s + (1.94 + 3.37i)10-s + (2.75 − 1.84i)11-s + (0.662 + 0.382i)12-s + 1.81·13-s + 2.98·15-s + (−0.5 + 0.866i)16-s + (−3.03 − 5.26i)17-s + (−2.09 + 1.20i)18-s + (−3.41 + 5.91i)19-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (0.382 − 0.220i)3-s + (0.249 + 0.433i)4-s + (1.51 + 0.871i)5-s + 0.312·6-s + 0.353i·8-s + (−0.402 + 0.696i)9-s + (0.616 + 1.06i)10-s + (0.831 − 0.555i)11-s + (0.191 + 0.110i)12-s + 0.504·13-s + 0.770·15-s + (−0.125 + 0.216i)16-s + (−0.736 − 1.27i)17-s + (−0.492 + 0.284i)18-s + (−0.783 + 1.35i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.526 - 0.850i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.526 - 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1078\)    =    \(2 \cdot 7^{2} \cdot 11\)
Sign: $0.526 - 0.850i$
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: \((2,\ 1078,\ (\ :1/2),\ 0.526 - 0.850i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.337561255\)
\(L(\frac12)\) \(\approx\) \(3.337561255\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.75 + 1.84i)T \)
good3 \( 1 + (-0.662 + 0.382i)T + (1.5 - 2.59i)T^{2} \)
5 \( 1 + (-3.37 - 1.94i)T + (2.5 + 4.33i)T^{2} \)
13 \( 1 - 1.81T + 13T^{2} \)
17 \( 1 + (3.03 + 5.26i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.41 - 5.91i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.18 + 3.79i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 9.36iT - 29T^{2} \)
31 \( 1 + (6.82 - 3.94i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2.30 + 3.99i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 3.58T + 41T^{2} \)
43 \( 1 - 3.25iT - 43T^{2} \)
47 \( 1 + (-0.176 - 0.101i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (2.49 + 4.31i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (4.50 - 2.59i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4.49 + 7.78i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-4.01 - 6.95i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 8.86T + 71T^{2} \)
73 \( 1 + (-0.0278 - 0.0482i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (7.34 + 4.24i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 - 10.4T + 83T^{2} \)
89 \( 1 + (1.46 + 0.843i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + 9.23iT - 97T^{2} \)
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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

−10.00376338742648718182692994700, −9.112618540621463815429884042837, −8.370315843632515554124495758118, −7.29378924056488696354681728614, −6.41912273890181938818108881081, −5.97825995537203500367415738026, −5.01794212194283976815384107281, −3.69068389082145050034966236866, −2.63378606639801375284434788726, −1.89666751441393549052870676234, 1.34750892424918315901896969414, 2.22081691602810806149870109497, 3.53126465548856139943434727924, 4.48152692002793566109467203546, 5.38582026130882252810192337751, 6.24930598313241621181390759402, 6.82487096193128158766318682747, 8.577602737644789961638704429479, 9.094649997392289715888825769458, 9.535260155882159332216369740182

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