Properties

Label 2-1078-7.2-c1-0-30
Degree $2$
Conductor $1078$
Sign $-0.701 - 0.712i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.701 - 0.712i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.701 - 0.712i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1078\)    =    \(2 \cdot 7^{2} \cdot 11\)
Sign: $-0.701 - 0.712i$
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: \((2,\ 1078,\ (\ :1/2),\ -0.701 - 0.712i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3364929733\)
\(L(\frac12)\) \(\approx\) \(0.3364929733\)
\(L(\frac{3}{2})\) 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

−9.121148476182940348980395001732, −8.409838256020649654074851453109, −7.51484096418332555888494429577, −6.93301970404005469022236361331, −5.83983371739778540394085798278, −5.53680126006469395160874514065, −4.61949779704998171692916017177, −2.27996233513956075957863307642, −1.32087215159018707272653887482, −0.19400707111230236578173742123, 2.21429530804320868089054660550, 3.30514515412560041918856082066, 4.21979447396846498572077054378, 5.08192896163729125521252635361, 6.21904175356128230193433121938, 6.66793935507070859952216502062, 8.223613708405864440027302258673, 9.205895077930689392012152634891, 10.02526220651364162556211563096, 10.24360798334172985222726614126

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