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Properties

Label 1-709-709.76-r0-0-0
Degree $1$
Conductor $709$
Sign $-0.351 - 0.936i$
Analytic cond. $3.29258$
Root an. cond. $3.29258$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.887 − 0.461i)2-s + (0.734 − 0.678i)3-s + (0.574 − 0.818i)4-s + (−0.931 + 0.364i)5-s + (0.339 − 0.940i)6-s + (0.977 + 0.211i)7-s + (0.132 − 0.991i)8-s + (0.0797 − 0.996i)9-s + (−0.658 + 0.752i)10-s + (0.802 − 0.596i)11-s + (−0.132 − 0.991i)12-s + (−0.977 + 0.211i)13-s + (0.964 − 0.263i)14-s + (−0.437 + 0.899i)15-s + (−0.339 − 0.940i)16-s + (−0.802 − 0.596i)17-s + ⋯
L(s)  = 1  + (0.887 − 0.461i)2-s + (0.734 − 0.678i)3-s + (0.574 − 0.818i)4-s + (−0.931 + 0.364i)5-s + (0.339 − 0.940i)6-s + (0.977 + 0.211i)7-s + (0.132 − 0.991i)8-s + (0.0797 − 0.996i)9-s + (−0.658 + 0.752i)10-s + (0.802 − 0.596i)11-s + (−0.132 − 0.991i)12-s + (−0.977 + 0.211i)13-s + (0.964 − 0.263i)14-s + (−0.437 + 0.899i)15-s + (−0.339 − 0.940i)16-s + (−0.802 − 0.596i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 709 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.351 - 0.936i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 709 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.351 - 0.936i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(709\)
Sign: $-0.351 - 0.936i$
Analytic conductor: \(3.29258\)
Root analytic conductor: \(3.29258\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{709} (76, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 709,\ (0:\ ),\ -0.351 - 0.936i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.657090115 - 2.391030646i\)
\(L(\frac12)\) \(\approx\) \(1.657090115 - 2.391030646i\)
\(L(1)\) \(\approx\) \(1.696729010 - 1.145737333i\)
\(L(1)\) \(\approx\) \(1.696729010 - 1.145737333i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad709 \( 1 \)
good2 \( 1 + (0.887 - 0.461i)T \)
3 \( 1 + (0.734 - 0.678i)T \)
5 \( 1 + (-0.931 + 0.364i)T \)
7 \( 1 + (0.977 + 0.211i)T \)
11 \( 1 + (0.802 - 0.596i)T \)
13 \( 1 + (-0.977 + 0.211i)T \)
17 \( 1 + (-0.802 - 0.596i)T \)
19 \( 1 + (-0.132 + 0.991i)T \)
23 \( 1 + (0.987 + 0.159i)T \)
29 \( 1 + (0.388 - 0.921i)T \)
31 \( 1 + (0.437 - 0.899i)T \)
37 \( 1 + (-0.977 - 0.211i)T \)
41 \( 1 + (0.697 + 0.716i)T \)
43 \( 1 + (0.185 + 0.982i)T \)
47 \( 1 + (-0.987 - 0.159i)T \)
53 \( 1 + (0.437 - 0.899i)T \)
59 \( 1 + (-0.132 + 0.991i)T \)
61 \( 1 + (-0.288 + 0.957i)T \)
67 \( 1 + (0.994 - 0.106i)T \)
71 \( 1 + (-0.658 - 0.752i)T \)
73 \( 1 + (-0.949 - 0.314i)T \)
79 \( 1 + (0.530 + 0.847i)T \)
83 \( 1 + (-0.910 + 0.413i)T \)
89 \( 1 + (0.769 + 0.638i)T \)
97 \( 1 + (0.237 + 0.971i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−22.82467210712002991922061639513, −21.99288722534696225754532379952, −21.37413260869872142207712088772, −20.35342307295081581489707578708, −20.00205894233277942825280959001, −19.23087260945072583881083612421, −17.41066940658719949007609009574, −17.14511312516270585529307215917, −15.9083635673858806581406149447, −15.33077678023807735325968492711, −14.69313851761461111520192940118, −14.128747853036350430137887811073, −12.98230747986923713683920736801, −12.20349956355234072536364537432, −11.26881238532095987327307873644, −10.54858556103061033753491130134, −8.94338112478429939240900530980, −8.51598984212056145586473834869, −7.42074563695229243274256077375, −6.91784977637205500909362796721, −5.01541262302124542169847627358, −4.74030300872069631145123557806, −3.94299588731178510877076255848, −2.93492417879433943760652425936, −1.781212899572368355619723489210, 0.99349481004275638070096404663, 2.15525763140248552841726041133, 2.9693825983509943874478187533, 3.99273080084320120895209632694, 4.72662671643946998462993288509, 6.10712442536409563232017940289, 7.00215099891872463509479243880, 7.7663798258218740945163051358, 8.72574336787603201782523985890, 9.79029902859646170747569729193, 11.13062966054382953099597682328, 11.698251041353969918679402163685, 12.2472650626851350420136039110, 13.34233744974037924055181087403, 14.18816895622381570566792021916, 14.77478415544140630050117416773, 15.248032897960959224104119260408, 16.44237492635884225031103970294, 17.700858322339596990468176632361, 18.70472531103824712536586556724, 19.34768998965605239255366574597, 19.79863797670135204831660604711, 20.81672608583749795619282842163, 21.363728557146327186023456979804, 22.55117199322577436296130009362

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