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Properties

Label 2-1078-77.54-c1-0-26
Degree $2$
Conductor $1078$
Sign $0.898 + 0.438i$
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 + (1.60 − 0.923i)3-s + (0.499 + 0.866i)4-s + (2.14 + 1.23i)5-s − 1.84·6-s − 0.999i·8-s + (0.207 − 0.358i)9-s + (−1.23 − 2.14i)10-s + (3.29 + 0.397i)11-s + (1.60 + 0.923i)12-s + 1.96·13-s + 4.56·15-s + (−0.5 + 0.866i)16-s + (−2.82 − 4.89i)17-s + (−0.358 + 0.207i)18-s + (−0.453 + 0.785i)19-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (0.923 − 0.533i)3-s + (0.249 + 0.433i)4-s + (0.957 + 0.552i)5-s − 0.754·6-s − 0.353i·8-s + (0.0690 − 0.119i)9-s + (−0.390 − 0.676i)10-s + (0.992 + 0.119i)11-s + (0.461 + 0.266i)12-s + 0.544·13-s + 1.17·15-s + (−0.125 + 0.216i)16-s + (−0.685 − 1.18i)17-s + (−0.0845 + 0.0488i)18-s + (−0.104 + 0.180i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.898 + 0.438i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.898 + 0.438i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1078\)    =    \(2 \cdot 7^{2} \cdot 11\)
Sign: $0.898 + 0.438i$
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: \((2,\ 1078,\ (\ :1/2),\ 0.898 + 0.438i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.062667077\)
\(L(\frac12)\) \(\approx\) \(2.062667077\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.29 - 0.397i)T \)
good3 \( 1 + (-1.60 + 0.923i)T + (1.5 - 2.59i)T^{2} \)
5 \( 1 + (-2.14 - 1.23i)T + (2.5 + 4.33i)T^{2} \)
13 \( 1 - 1.96T + 13T^{2} \)
17 \( 1 + (2.82 + 4.89i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.453 - 0.785i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.468 + 0.811i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 7.50iT - 29T^{2} \)
31 \( 1 + (-4.08 + 2.35i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-4.81 + 8.33i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 6.93T + 41T^{2} \)
43 \( 1 - 6.80iT - 43T^{2} \)
47 \( 1 + (-3.46 - 2.00i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (3.28 + 5.68i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (0.836 - 0.482i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (5.95 - 10.3i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.36 + 5.81i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 11.5T + 71T^{2} \)
73 \( 1 + (-5.43 - 9.40i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (7.34 + 4.24i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 - 4.83T + 83T^{2} \)
89 \( 1 + (-11.0 - 6.36i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 - 3.82iT - 97T^{2} \)
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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

−9.428369194239260113607479707517, −9.230233753698514219255821037147, −8.322165493610217449235563818966, −7.37813173307016002200585805952, −6.71487097830360865874214234738, −5.83872025463066850263501726303, −4.33282976290204460294238976040, −3.03180798064399320671490377045, −2.35989895091360052359372902633, −1.33880244716976928527739680150, 1.27964255845316913314212405587, 2.40668357255789396167729118649, 3.73837290670565168024979094072, 4.63838549936681673536396435291, 6.09589009276488888557025171550, 6.30063976712402324342506381096, 7.76596201982012646529990138579, 8.639407476131481907788553708163, 9.011367793368713069407848888803, 9.668848739051583876785902442562

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