Properties

Label 2-1078-7.2-c1-0-7
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 + (1.61 + 2.80i)3-s + (−0.499 − 0.866i)4-s + (−1.61 + 2.80i)5-s − 3.23·6-s + 0.999·8-s + (−3.73 + 6.47i)9-s + (−1.61 − 2.80i)10-s + (−0.5 − 0.866i)11-s + (1.61 − 2.80i)12-s + 1.23·13-s − 10.4·15-s + (−0.5 + 0.866i)16-s + (3.23 + 5.60i)17-s + (−3.73 − 6.47i)18-s + (1.38 − 2.39i)19-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (0.934 + 1.61i)3-s + (−0.249 − 0.433i)4-s + (−0.723 + 1.25i)5-s − 1.32·6-s + 0.353·8-s + (−1.24 + 2.15i)9-s + (−0.511 − 0.886i)10-s + (−0.150 − 0.261i)11-s + (0.467 − 0.809i)12-s + 0.342·13-s − 2.70·15-s + (−0.125 + 0.216i)16-s + (0.784 + 1.35i)17-s + (−0.880 − 1.52i)18-s + (0.317 − 0.549i)19-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

\(L(1)\) \(\approx\) \(1.320687618\)
\(L(\frac12)\) \(\approx\) \(1.320687618\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.61 - 2.80i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (1.61 - 2.80i)T + (-2.5 - 4.33i)T^{2} \)
13 \( 1 - 1.23T + 13T^{2} \)
17 \( 1 + (-3.23 - 5.60i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.38 + 2.39i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (2 - 3.46i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 4.47T + 29T^{2} \)
31 \( 1 + (1 + 1.73i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-5.47 + 9.47i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 6.47T + 41T^{2} \)
43 \( 1 + 1.52T + 43T^{2} \)
47 \( 1 + (-1 + 1.73i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-0.236 - 0.408i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (3.61 + 6.26i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.61 + 4.53i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-7.70 - 13.3i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 2.47T + 71T^{2} \)
73 \( 1 + (-2.47 - 4.28i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 10.1T + 83T^{2} \)
89 \( 1 + (5 - 8.66i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 3.52T + 97T^{2} \)
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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

−10.25112248338798068207408983133, −9.579410063564343782191275243783, −8.801855244583854520503490494316, −7.900450737225095132405297369434, −7.51148348895111774359892507744, −6.13561696880837170244503892386, −5.27911606995012011257866548434, −3.87936246501667235564082525803, −3.67350723487970640813544777321, −2.45247768376633488459939643326, 0.62274468191681132681374089533, 1.47440520033644363541171751711, 2.68056070979100287332154591508, 3.64336587891415512283406583339, 4.83929534306983776112955265874, 6.11971204866869489111216787781, 7.36816339226097863728342080878, 7.78628572518744370547544682960, 8.457789018410234657318521314520, 9.109875962892410110040575523730

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