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Properties

Label 2-1078-7.4-c1-0-30
Degree $2$
Conductor $1078$
Sign $-0.900 - 0.435i$
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 + (1.41 − 2.44i)3-s + (−0.499 + 0.866i)4-s − 2.82·6-s + 0.999·8-s + (−2.49 − 4.33i)9-s + (0.5 − 0.866i)11-s + (1.41 + 2.44i)12-s − 4.24·13-s + (−0.5 − 0.866i)16-s + (−1.41 + 2.44i)17-s + (−2.5 + 4.33i)18-s + (−2.12 − 3.67i)19-s − 0.999·22-s + (−3 − 5.19i)23-s + (1.41 − 2.44i)24-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (0.816 − 1.41i)3-s + (−0.249 + 0.433i)4-s − 1.15·6-s + 0.353·8-s + (−0.833 − 1.44i)9-s + (0.150 − 0.261i)11-s + (0.408 + 0.707i)12-s − 1.17·13-s + (−0.125 − 0.216i)16-s + (−0.342 + 0.594i)17-s + (−0.589 + 1.02i)18-s + (−0.486 − 0.842i)19-s − 0.213·22-s + (−0.625 − 1.08i)23-s + (0.288 − 0.499i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.900 - 0.435i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.900 - 0.435i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1078\)    =    \(2 \cdot 7^{2} \cdot 11\)
Sign: $-0.900 - 0.435i$
Analytic conductor: \(8.60787\)
Root analytic conductor: \(2.93391\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1078} (67, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1078,\ (\ :1/2),\ -0.900 - 0.435i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.120151156\)
\(L(\frac12)\) \(\approx\) \(1.120151156\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.41 + 2.44i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (-2.5 + 4.33i)T^{2} \)
13 \( 1 + 4.24T + 13T^{2} \)
17 \( 1 + (1.41 - 2.44i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.12 + 3.67i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (3 + 5.19i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 4T + 29T^{2} \)
31 \( 1 + (3.53 - 6.12i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (1 + 1.73i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 2.82T + 41T^{2} \)
43 \( 1 - 10T + 43T^{2} \)
47 \( 1 + (6.36 + 11.0i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (1 - 1.73i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-5.65 + 9.79i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (4.94 + 8.57i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (4 - 6.92i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 16T + 71T^{2} \)
73 \( 1 + (4.24 - 7.34i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-4 - 6.92i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 12.7T + 83T^{2} \)
89 \( 1 + (-3.53 - 6.12i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 7.07T + 97T^{2} \)
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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

−9.171316356343740830047405349700, −8.558859488221935065722470645786, −7.919059253841763668079832475053, −7.00373500656771295472481831342, −6.45319301036509355692565872599, −4.96062900121633490672830171472, −3.70046234650284048089165966192, −2.52574081106356780936833481250, −1.97992678074676154169877537150, −0.47469420424988958869217618089, 2.11040485851714521394313151517, 3.38027428860066129907985008354, 4.32104686664175196371116493131, 5.06540838151111309884663631527, 6.00618605548551538378192941071, 7.39702067397034029106159043892, 7.83414443072269252509386446566, 9.037594107898982837976759381387, 9.376942397849247572567098136008, 10.00942849901404432893968260493

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