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Properties

Label 2-1078-77.10-c1-0-21
Degree $2$
Conductor $1078$
Sign $-0.0479 + 0.998i$
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 + (1.60 + 0.923i)3-s + (0.499 − 0.866i)4-s + (−2.80 + 1.61i)5-s − 1.84·6-s + 0.999i·8-s + (0.207 + 0.358i)9-s + (1.61 − 2.80i)10-s + (−1.23 + 3.07i)11-s + (1.60 − 0.923i)12-s − 6.10·13-s − 5.98·15-s + (−0.5 − 0.866i)16-s + (4.06 − 7.04i)17-s + (−0.358 − 0.207i)18-s + (−3.30 − 5.72i)19-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (0.923 + 0.533i)3-s + (0.249 − 0.433i)4-s + (−1.25 + 0.723i)5-s − 0.754·6-s + 0.353i·8-s + (0.0690 + 0.119i)9-s + (0.511 − 0.886i)10-s + (−0.372 + 0.927i)11-s + (0.461 − 0.266i)12-s − 1.69·13-s − 1.54·15-s + (−0.125 − 0.216i)16-s + (0.986 − 1.70i)17-s + (−0.0845 − 0.0488i)18-s + (−0.758 − 1.31i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0479 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0479 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1078\)    =    \(2 \cdot 7^{2} \cdot 11\)
Sign: $-0.0479 + 0.998i$
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: \((2,\ 1078,\ (\ :1/2),\ -0.0479 + 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2338230473\)
\(L(\frac12)\) \(\approx\) \(0.2338230473\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.23 - 3.07i)T \)
good3 \( 1 + (-1.60 - 0.923i)T + (1.5 + 2.59i)T^{2} \)
5 \( 1 + (2.80 - 1.61i)T + (2.5 - 4.33i)T^{2} \)
13 \( 1 + 6.10T + 13T^{2} \)
17 \( 1 + (-4.06 + 7.04i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.30 + 5.72i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2.65 - 4.59i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 1.32iT - 29T^{2} \)
31 \( 1 + (-2.03 - 1.17i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (2.64 + 4.57i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 1.22T + 41T^{2} \)
43 \( 1 - 3.73iT - 43T^{2} \)
47 \( 1 + (1.47 - 0.853i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-1.99 + 3.44i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (7.82 + 4.51i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (4.27 + 7.41i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.17 - 2.03i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 7.17T + 71T^{2} \)
73 \( 1 + (5.49 - 9.51i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (7.34 - 4.24i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + 5.60T + 83T^{2} \)
89 \( 1 + (-4.03 + 2.33i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 3.82iT - 97T^{2} \)
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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

−9.614979018083477531092365500878, −8.941768682602161208789985664141, −7.80114171590168104197720096387, −7.41454685043958849068237030971, −6.85113621565030228168049592717, −5.13471613898822354393707493380, −4.41294332270320250665980129537, −3.09454248253222668059828095815, −2.54537402275618926927450631933, −0.11162984105040232218714145664, 1.47252044132091306366831155134, 2.73977976798393516786393396490, 3.62784049868317650608160360263, 4.64267564235210067483491921903, 5.94467159517221593574897450015, 7.32888297536161224282729669573, 7.88456814855701435270935344357, 8.401171339862733670421997638479, 8.856242593587960732581197749082, 10.17320384163914191867267176133

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