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Properties

Label 2-1078-77.54-c1-0-1
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: $\chi_{1078} (901, \cdot )$
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

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

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