Properties

Label 2-1078-77.54-c1-0-38
Degree $2$
Conductor $1078$
Sign $0.874 + 0.485i$
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 + (2.68 − 1.54i)3-s + (0.499 + 0.866i)4-s + (−0.559 − 0.323i)5-s + 3.09·6-s + 0.999i·8-s + (3.29 − 5.70i)9-s + (−0.323 − 0.559i)10-s + (0.155 − 3.31i)11-s + (2.68 + 1.54i)12-s + 3.09·13-s − 2·15-s + (−0.5 + 0.866i)16-s + (1.87 + 3.24i)17-s + (5.70 − 3.29i)18-s + (−2.77 + 4.80i)19-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (1.54 − 0.893i)3-s + (0.249 + 0.433i)4-s + (−0.250 − 0.144i)5-s + 1.26·6-s + 0.353i·8-s + (1.09 − 1.90i)9-s + (−0.102 − 0.176i)10-s + (0.0469 − 0.998i)11-s + (0.773 + 0.446i)12-s + 0.858·13-s − 0.516·15-s + (−0.125 + 0.216i)16-s + (0.453 + 0.785i)17-s + (1.34 − 0.775i)18-s + (−0.636 + 1.10i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.874 + 0.485i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.874 + 0.485i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1078\)    =    \(2 \cdot 7^{2} \cdot 11\)
Sign: $0.874 + 0.485i$
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: \((2,\ 1078,\ (\ :1/2),\ 0.874 + 0.485i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.775833281\)
\(L(\frac12)\) \(\approx\) \(3.775833281\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.155 + 3.31i)T \)
good3 \( 1 + (-2.68 + 1.54i)T + (1.5 - 2.59i)T^{2} \)
5 \( 1 + (0.559 + 0.323i)T + (2.5 + 4.33i)T^{2} \)
13 \( 1 - 3.09T + 13T^{2} \)
17 \( 1 + (-1.87 - 3.24i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.77 - 4.80i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (2 - 3.46i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 7.58iT - 29T^{2} \)
31 \( 1 + (-1.00 + 0.578i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2.79 + 4.83i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 5.03T + 41T^{2} \)
43 \( 1 - 11.1iT - 43T^{2} \)
47 \( 1 + (-4.35 - 2.51i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-1.20 - 2.09i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (2.68 - 1.54i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (4.64 - 8.04i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.791 + 1.37i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 2T + 71T^{2} \)
73 \( 1 + (3.16 + 5.47i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-3.46 - 2i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + 9.15T + 83T^{2} \)
89 \( 1 + (-8.48 - 4.89i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 - 15.9iT - 97T^{2} \)
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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

−9.520135886579250276874146407736, −8.575687773488105642121258917044, −8.036124824769731158017550837659, −7.66619949431067534503459220927, −6.26542130751195944471606183630, −5.98892762747814965254777064138, −4.08538397628216379607448401524, −3.63900087634715470872365952511, −2.57027373574227079543277666982, −1.39001034914917193001275771208, 1.86501967346130582687233585596, 2.87342169833994245746142139694, 3.64492282812155339728541527906, 4.46140953584934087795287247892, 5.19961945154603282985281571111, 6.75074428220913036291869601395, 7.50380831975881303281509565886, 8.560738944725353081389077949367, 9.118477158173706484872838062134, 9.983508556064650599759231327591

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