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Properties

Label 2-1078-77.54-c1-0-18
Degree $2$
Conductor $1078$
Sign $0.228 + 0.973i$
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 + (−1.77 − 1.02i)5-s − 0.765·6-s − 0.999i·8-s + (−1.20 + 2.09i)9-s + (1.02 + 1.77i)10-s + (0.649 − 3.25i)11-s + (0.662 + 0.382i)12-s + 6.59·13-s − 1.57·15-s + (−0.5 + 0.866i)16-s + (1.80 + 3.12i)17-s + (2.09 − 1.20i)18-s + (0.439 − 0.760i)19-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (0.382 − 0.220i)3-s + (0.249 + 0.433i)4-s + (−0.794 − 0.458i)5-s − 0.312·6-s − 0.353i·8-s + (−0.402 + 0.696i)9-s + (0.324 + 0.561i)10-s + (0.195 − 0.980i)11-s + (0.191 + 0.110i)12-s + 1.82·13-s − 0.405·15-s + (−0.125 + 0.216i)16-s + (0.437 + 0.758i)17-s + (0.492 − 0.284i)18-s + (0.100 − 0.174i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.228 + 0.973i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.228 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1078\)    =    \(2 \cdot 7^{2} \cdot 11\)
Sign: $0.228 + 0.973i$
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: \((2,\ 1078,\ (\ :1/2),\ 0.228 + 0.973i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.163187295\)
\(L(\frac12)\) \(\approx\) \(1.163187295\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.649 + 3.25i)T \)
good3 \( 1 + (-0.662 + 0.382i)T + (1.5 - 2.59i)T^{2} \)
5 \( 1 + (1.77 + 1.02i)T + (2.5 + 4.33i)T^{2} \)
13 \( 1 - 6.59T + 13T^{2} \)
17 \( 1 + (-1.80 - 3.12i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.439 + 0.760i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (3.30 - 5.73i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 6.18iT - 29T^{2} \)
31 \( 1 + (-5.61 + 3.24i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-5.52 + 9.56i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 2.36T + 41T^{2} \)
43 \( 1 + 7.81iT - 43T^{2} \)
47 \( 1 + (4.97 + 2.87i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (0.214 + 0.372i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-2.78 + 1.60i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-5.66 + 9.81i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.48 + 2.56i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 2.13T + 71T^{2} \)
73 \( 1 + (-5.41 - 9.37i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-7.34 - 4.24i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 - 12.3T + 83T^{2} \)
89 \( 1 + (-5.82 - 3.36i)T + (44.5 + 77.0i)T^{2} \)
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

−9.579458689646975288511959631488, −8.607730201579285728353355170491, −8.187894965547120982919153892766, −7.74493557378877536869785474142, −6.33233275161491531977065902017, −5.57201127610251757775173290617, −3.98069406149053669931689430092, −3.48542748128455362048683828607, −2.05652394654472832075623041072, −0.74299048224009907660657037755, 1.16744746796294225126946326649, 2.88588414331074555981203451571, 3.73308098784022180152216576184, 4.78549082010528559750110913782, 6.22992868983794642803970035804, 6.67217558902041504468802898724, 7.77950261181211744665208457016, 8.374444725167270694225486702532, 9.121522784441999075194258794800, 9.948935123518394732904992047728

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