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Properties

Label 2-1078-77.10-c1-0-9
Degree $2$
Conductor $1078$
Sign $0.913 - 0.407i$
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 + (−3.37 + 1.94i)5-s + 0.765·6-s + 0.999i·8-s + (−1.20 − 2.09i)9-s + (1.94 − 3.37i)10-s + (−2.97 − 1.46i)11-s + (−0.662 + 0.382i)12-s + 1.81·13-s + 2.98·15-s + (−0.5 − 0.866i)16-s + (−3.03 + 5.26i)17-s + (2.09 + 1.20i)18-s + (−3.41 − 5.91i)19-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (−0.382 − 0.220i)3-s + (0.249 − 0.433i)4-s + (−1.51 + 0.871i)5-s + 0.312·6-s + 0.353i·8-s + (−0.402 − 0.696i)9-s + (0.616 − 1.06i)10-s + (−0.896 − 0.442i)11-s + (−0.191 + 0.110i)12-s + 0.504·13-s + 0.770·15-s + (−0.125 − 0.216i)16-s + (−0.736 + 1.27i)17-s + (0.492 + 0.284i)18-s + (−0.783 − 1.35i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.913 - 0.407i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.913 - 0.407i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1078\)    =    \(2 \cdot 7^{2} \cdot 11\)
Sign: $0.913 - 0.407i$
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: \((2,\ 1078,\ (\ :1/2),\ 0.913 - 0.407i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4827970660\)
\(L(\frac12)\) \(\approx\) \(0.4827970660\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.97 + 1.46i)T \)
good3 \( 1 + (0.662 + 0.382i)T + (1.5 + 2.59i)T^{2} \)
5 \( 1 + (3.37 - 1.94i)T + (2.5 - 4.33i)T^{2} \)
13 \( 1 - 1.81T + 13T^{2} \)
17 \( 1 + (3.03 - 5.26i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.41 + 5.91i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2.18 - 3.79i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 9.36iT - 29T^{2} \)
31 \( 1 + (-6.82 - 3.94i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-2.30 - 3.99i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 3.58T + 41T^{2} \)
43 \( 1 - 3.25iT - 43T^{2} \)
47 \( 1 + (0.176 - 0.101i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (2.49 - 4.31i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-4.50 - 2.59i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4.49 - 7.78i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-4.01 + 6.95i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 8.86T + 71T^{2} \)
73 \( 1 + (-0.0278 + 0.0482i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-7.34 + 4.24i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 - 10.4T + 83T^{2} \)
89 \( 1 + (-1.46 + 0.843i)T + (44.5 - 77.0i)T^{2} \)
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

−10.06601881059316970329849271203, −8.754253840733844515953330035934, −8.285874754994530779090022153644, −7.49804204608021184062194150779, −6.55764042107317516531535783054, −6.16779593467469290010857782229, −4.71909348621203092403129977087, −3.63389979434554026985783686010, −2.64047028575001050303332540841, −0.57900552592140184877073403687, 0.55824924974210716314609477957, 2.30944781096178042754239643007, 3.61441326566017196195932439951, 4.62816884335118643372963693716, 5.20958879419509124727789932195, 6.68713606726854147461636224657, 7.67248615258938562103842314757, 8.274663893009955888753049358320, 8.771247444874386407107806293027, 9.910998897524103800544323816159

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