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Properties

Label 2-1078-7.4-c1-0-29
Degree $2$
Conductor $1078$
Sign $-0.386 + 0.922i$
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 + (1.32 − 2.29i)3-s + (−0.499 + 0.866i)4-s + (−0.822 − 1.42i)5-s + 2.64·6-s − 0.999·8-s + (−2 − 3.46i)9-s + (0.822 − 1.42i)10-s + (−0.5 + 0.866i)11-s + (1.32 + 2.29i)12-s − 5·13-s − 4.35·15-s + (−0.5 − 0.866i)16-s + (3 − 5.19i)17-s + (1.99 − 3.46i)18-s + (−2.82 − 4.88i)19-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (0.763 − 1.32i)3-s + (−0.249 + 0.433i)4-s + (−0.368 − 0.637i)5-s + 1.08·6-s − 0.353·8-s + (−0.666 − 1.15i)9-s + (0.260 − 0.450i)10-s + (−0.150 + 0.261i)11-s + (0.381 + 0.661i)12-s − 1.38·13-s − 1.12·15-s + (−0.125 − 0.216i)16-s + (0.727 − 1.26i)17-s + (0.471 − 0.816i)18-s + (−0.647 − 1.12i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.386 + 0.922i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.386 + 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1078\)    =    \(2 \cdot 7^{2} \cdot 11\)
Sign: $-0.386 + 0.922i$
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: \((2,\ 1078,\ (\ :1/2),\ -0.386 + 0.922i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.685634290\)
\(L(\frac12)\) \(\approx\) \(1.685634290\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.32 + 2.29i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (0.822 + 1.42i)T + (-2.5 + 4.33i)T^{2} \)
13 \( 1 + 5T + 13T^{2} \)
17 \( 1 + (-3 + 5.19i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.82 + 4.88i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.822 + 1.42i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 6.29T + 29T^{2} \)
31 \( 1 + (2 - 3.46i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (1.82 + 3.15i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 10.9T + 41T^{2} \)
43 \( 1 + 4T + 43T^{2} \)
47 \( 1 + (-1.35 - 2.34i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (0.822 - 1.42i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-2.32 + 4.02i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-7.14 - 12.3i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-5.96 + 10.3i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 4.35T + 71T^{2} \)
73 \( 1 + (-0.177 + 0.306i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-1.32 - 2.29i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 2.70T + 83T^{2} \)
89 \( 1 + (-3.29 - 5.70i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 16.2T + 97T^{2} \)
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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

−9.237698215543252735082161023847, −8.502204012914601307573528818538, −7.88075160260779376241519720542, −7.04910239769335878197846050192, −6.71792122841608023526109343554, −5.18964317795485149413918683492, −4.62640196067203972810318152721, −3.09421587999297920405235900063, −2.24997757033258156038998580017, −0.58201281571905823276832992890, 2.07420813306571977359161571275, 3.23800913150907555667524250169, 3.69251203511804993708459772335, 4.66349942605139029108726604584, 5.52755961596187646416650777992, 6.75033189537099838005175777764, 7.998216170880631202034470997330, 8.550482229084694476455155501234, 9.713099943642066418054941842974, 10.17463747185570232267377293106

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