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Properties

Label 2-1078-77.10-c1-0-30
Degree $2$
Conductor $1078$
Sign $-0.357 + 0.933i$
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 + (−3.14 + 1.06i)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 + (−0.947 + 0.320i)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

\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.357 + 0.933i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.357 + 0.933i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1078\)    =    \(2 \cdot 7^{2} \cdot 11\)
Sign: $-0.357 + 0.933i$
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: \((2,\ 1078,\ (\ :1/2),\ -0.357 + 0.933i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.137622483\)
\(L(\frac12)\) \(\approx\) \(2.137622483\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.14 - 1.06i)T \)
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

−9.774080148462216776842050900712, −8.910574155915376617118927476839, −8.029964278743761704923420383545, −6.79758519993795739189240384338, −5.86164004647448628603522219382, −5.60356366254043525737724069516, −4.38166349523511581796462513000, −3.31253369441316338201808278224, −2.11852926680351253583531465417, −0.828850451186696781327976153563, 1.79453109743576569174745342190, 3.04386536272703912611255228772, 3.97490854024293319983405691875, 5.36616774383213937194036026896, 5.72813064738181180984791667582, 6.42151663883699400155515552104, 7.65538186558080485826710414213, 8.295567276861996039196036452268, 9.306672169524564665839882100525, 10.49596096722305249682846470986

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