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Properties

Label 2-1078-77.10-c1-0-11
Degree $2$
Conductor $1078$
Sign $0.958 - 0.286i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.958 - 0.286i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.958 - 0.286i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1078\)    =    \(2 \cdot 7^{2} \cdot 11\)
Sign: $0.958 - 0.286i$
Analytic conductor: \(8.60787\)
Root analytic conductor: \(2.93391\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1078} (1011, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1078,\ (\ :1/2),\ 0.958 - 0.286i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.406948955\)
\(L(\frac12)\) \(\approx\) \(1.406948955\)
\(L(\frac{3}{2})\) 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

−10.05897252759707095500207574718, −9.283759942495134624070875044269, −8.013097533618652509938048654131, −7.14028293739723283089273956000, −6.76439611720543919262384861636, −5.55918838254907614008635991290, −4.57813828679870082198888108897, −3.45338034278718435132718006855, −3.08444366820627596138098487271, −1.10885684859451346563612217306, 0.68036187880123629129468809227, 2.77672592247592585052915246223, 4.02691167307305212927269343698, 4.50190084498840467157234009089, 5.41086562050127405230688778801, 6.32451935352743894534513979248, 7.46884554277099681179269564557, 8.113966054921979678550699231083, 8.692159979398577548997445345485, 9.818061797033351496721294093319

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