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Properties

Label 2-1078-7.4-c1-0-2
Degree $2$
Conductor $1078$
Sign $-0.605 - 0.795i$
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 + (−0.618 + 1.07i)3-s + (−0.499 + 0.866i)4-s + (0.618 + 1.07i)5-s + 1.23·6-s + 0.999·8-s + (0.736 + 1.27i)9-s + (0.618 − 1.07i)10-s + (−0.5 + 0.866i)11-s + (−0.618 − 1.07i)12-s − 3.23·13-s − 1.52·15-s + (−0.5 − 0.866i)16-s + (−1.23 + 2.14i)17-s + (0.736 − 1.27i)18-s + (3.61 + 6.26i)19-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.356 + 0.618i)3-s + (−0.249 + 0.433i)4-s + (0.276 + 0.478i)5-s + 0.504·6-s + 0.353·8-s + (0.245 + 0.424i)9-s + (0.195 − 0.338i)10-s + (−0.150 + 0.261i)11-s + (−0.178 − 0.309i)12-s − 0.897·13-s − 0.394·15-s + (−0.125 − 0.216i)16-s + (−0.299 + 0.519i)17-s + (0.173 − 0.300i)18-s + (0.830 + 1.43i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.605 - 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.605 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1078\)    =    \(2 \cdot 7^{2} \cdot 11\)
Sign: $-0.605 - 0.795i$
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: \((2,\ 1078,\ (\ :1/2),\ -0.605 - 0.795i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7297516762\)
\(L(\frac12)\) \(\approx\) \(0.7297516762\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.618 - 1.07i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (-0.618 - 1.07i)T + (-2.5 + 4.33i)T^{2} \)
13 \( 1 + 3.23T + 13T^{2} \)
17 \( 1 + (1.23 - 2.14i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.61 - 6.26i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (2 + 3.46i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 4.47T + 29T^{2} \)
31 \( 1 + (1 - 1.73i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (3.47 + 6.01i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 2.47T + 41T^{2} \)
43 \( 1 + 10.4T + 43T^{2} \)
47 \( 1 + (-1 - 1.73i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (4.23 - 7.33i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (1.38 - 2.39i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.381 - 0.661i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (5.70 - 9.88i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 6.47T + 71T^{2} \)
73 \( 1 + (6.47 - 11.2i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 12.1T + 83T^{2} \)
89 \( 1 + (5 + 8.66i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 12.4T + 97T^{2} \)
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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

−10.25450152624391286981520704818, −9.777629152257218873133493258111, −8.641842707327175533572535548041, −7.79375237974230476953859144102, −6.93021602216735072285287830156, −5.79223650833497379561752628318, −4.83826726405224177056753981407, −4.00611796174435292129632030225, −2.80902468521012605189956680420, −1.74263471571307998272752780359, 0.38203445109774266797855121817, 1.64250698663615982986929583990, 3.17674722836156686213198933326, 4.78982086568281023286513943456, 5.29389836332510262909658975736, 6.44618252827781767675097793514, 7.02810085546022361708441714654, 7.75777791818938844357864008306, 8.798020719445499765508857434237, 9.488578901023015397418172445836

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