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Properties

Label 2-1078-7.2-c1-0-0
Degree $2$
Conductor $1078$
Sign $0.605 - 0.795i$
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  + (0.5 − 0.866i)2-s + (−1.41 − 2.44i)3-s + (−0.499 − 0.866i)4-s − 2.82·6-s − 0.999·8-s + (−2.49 + 4.33i)9-s + (0.5 + 0.866i)11-s + (−1.41 + 2.44i)12-s − 2.82·13-s + (−0.5 + 0.866i)16-s + (−1.41 − 2.44i)17-s + (2.5 + 4.33i)18-s + (−2.82 + 4.89i)19-s + 0.999·22-s + (−4 + 6.92i)23-s + (1.41 + 2.44i)24-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.816 − 1.41i)3-s + (−0.249 − 0.433i)4-s − 1.15·6-s − 0.353·8-s + (−0.833 + 1.44i)9-s + (0.150 + 0.261i)11-s + (−0.408 + 0.707i)12-s − 0.784·13-s + (−0.125 + 0.216i)16-s + (−0.342 − 0.594i)17-s + (0.589 + 1.02i)18-s + (−0.648 + 1.12i)19-s + 0.213·22-s + (−0.834 + 1.44i)23-s + (0.288 + 0.499i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 - 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.605 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1078\)    =    \(2 \cdot 7^{2} \cdot 11\)
Sign: $0.605 - 0.795i$
Analytic conductor: \(8.60787\)
Root analytic conductor: \(2.93391\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1078} (177, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1078,\ (\ :1/2),\ 0.605 - 0.795i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.1092447746\)
\(L(\frac12)\) \(\approx\) \(0.1092447746\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 + 0.866i)T \)
7 \( 1 \)
11 \( 1 + (-0.5 - 0.866i)T \)
good3 \( 1 + (1.41 + 2.44i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (-2.5 - 4.33i)T^{2} \)
13 \( 1 + 2.82T + 13T^{2} \)
17 \( 1 + (1.41 + 2.44i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.82 - 4.89i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (4 - 6.92i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 2T + 29T^{2} \)
31 \( 1 + (4.24 + 7.34i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (1 - 1.73i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 2.82T + 41T^{2} \)
43 \( 1 + 4T + 43T^{2} \)
47 \( 1 + (1.41 - 2.44i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (7 + 12.1i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-4.24 - 7.34i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4.24 + 7.34i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (2 + 3.46i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + (-7.07 - 12.2i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-8 + 13.8i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 16.9T + 83T^{2} \)
89 \( 1 + (-5.65 + 9.79i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 16.9T + 97T^{2} \)
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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

−10.10349340939237940958838324464, −9.392819748708543250924905918187, −8.092075461912852311223599316372, −7.40096314733021809358899633314, −6.57773011602430694480845442351, −5.74315837278532694236260444089, −5.00166182047363317271349280411, −3.72624920539799383657352794416, −2.29175645989964627028322746729, −1.48467329748147434752561864809, 0.04710092569861754637216279350, 2.69415360592330803773184840775, 4.00423336700955718926099319130, 4.59754567637639164690481122170, 5.29534829355306092991459481824, 6.29138628633524247822797093318, 6.88039237464849577073921466002, 8.301175308246399173187728521221, 8.927829745097422253695022388149, 9.837399838119030344755740046830

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