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Properties

Label 2-1078-7.2-c1-0-2
Degree $2$
Conductor $1078$
Sign $-0.701 - 0.712i$
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 + (−0.499 − 0.866i)4-s + (−2 + 3.46i)5-s − 0.999·8-s + (1.5 − 2.59i)9-s + (1.99 + 3.46i)10-s + (0.5 + 0.866i)11-s − 2·13-s + (−0.5 + 0.866i)16-s + (−2 − 3.46i)17-s + (−1.5 − 2.59i)18-s + (−3 + 5.19i)19-s + 3.99·20-s + 0.999·22-s + (−2 + 3.46i)23-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.249 − 0.433i)4-s + (−0.894 + 1.54i)5-s − 0.353·8-s + (0.5 − 0.866i)9-s + (0.632 + 1.09i)10-s + (0.150 + 0.261i)11-s − 0.554·13-s + (−0.125 + 0.216i)16-s + (−0.485 − 0.840i)17-s + (−0.353 − 0.612i)18-s + (−0.688 + 1.19i)19-s + 0.894·20-s + 0.213·22-s + (−0.417 + 0.722i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.701 - 0.712i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.701 - 0.712i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1078\)    =    \(2 \cdot 7^{2} \cdot 11\)
Sign: $-0.701 - 0.712i$
Analytic conductor: \(8.60787\)
Root analytic conductor: \(2.93391\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1078} (177, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1078,\ (\ :1/2),\ -0.701 - 0.712i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3877161898\)
\(L(\frac12)\) \(\approx\) \(0.3877161898\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (2 - 3.46i)T + (-2.5 - 4.33i)T^{2} \)
13 \( 1 + 2T + 13T^{2} \)
17 \( 1 + (2 + 3.46i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (3 - 5.19i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (2 - 3.46i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 2T + 29T^{2} \)
31 \( 1 + (1 + 1.73i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (5 - 8.66i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 4T + 41T^{2} \)
43 \( 1 + 8T + 43T^{2} \)
47 \( 1 + (-1 + 1.73i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (3 + 5.19i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (6 + 10.3i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (7 - 12.1i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-6 - 10.3i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 8T + 71T^{2} \)
73 \( 1 + (-2 - 3.46i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 6T + 83T^{2} \)
89 \( 1 + (3 - 5.19i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 14T + 97T^{2} \)
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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

−10.08000033787343492297607970989, −9.875038947519482512342610663379, −8.559011360012276137600174435808, −7.48860344856437729135364096388, −6.85635868816809327932155916651, −6.11054143141054975237765970775, −4.69961615649638562593921852332, −3.73793485382853907577824326267, −3.18515153612322293879083860020, −1.92952037325567972179124686277, 0.14636214555811217611461498834, 1.92525314806828388113684110865, 3.66503930610902484547606492912, 4.68253937992556196726861317861, 4.86054059875307597403338837200, 6.11518121962573745494109512909, 7.22680704622501398107131237885, 7.905878418643774710519691748612, 8.675465696989419823372271681782, 9.133009965674375628519104596176

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