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Properties

Label 2-1716-11.3-c1-0-5
Degree $2$
Conductor $1716$
Sign $-0.574 - 0.818i$
Analytic cond. $13.7023$
Root an. cond. $3.70166$
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  + (−0.309 − 0.951i)3-s + (3.52 + 2.55i)5-s + (−1.38 + 4.25i)7-s + (−0.809 + 0.587i)9-s + (−3.14 + 1.04i)11-s + (−0.809 + 0.587i)13-s + (1.34 − 4.13i)15-s + (4.08 + 2.96i)17-s + (−1.58 − 4.87i)19-s + 4.46·21-s − 8.53·23-s + (4.30 + 13.2i)25-s + (0.809 + 0.587i)27-s + (0.262 − 0.806i)29-s + (−0.222 + 0.161i)31-s + ⋯
L(s)  = 1  + (−0.178 − 0.549i)3-s + (1.57 + 1.14i)5-s + (−0.521 + 1.60i)7-s + (−0.269 + 0.195i)9-s + (−0.948 + 0.315i)11-s + (−0.224 + 0.163i)13-s + (0.347 − 1.06i)15-s + (0.991 + 0.720i)17-s + (−0.363 − 1.11i)19-s + 0.975·21-s − 1.78·23-s + (0.861 + 2.65i)25-s + (0.155 + 0.113i)27-s + (0.0486 − 0.149i)29-s + (−0.0400 + 0.0290i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1716 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.574 - 0.818i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1716 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.574 - 0.818i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1716\)    =    \(2^{2} \cdot 3 \cdot 11 \cdot 13\)
Sign: $-0.574 - 0.818i$
Analytic conductor: \(13.7023\)
Root analytic conductor: \(3.70166\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1716} (157, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1716,\ (\ :1/2),\ -0.574 - 0.818i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.401973164\)
\(L(\frac12)\) \(\approx\) \(1.401973164\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (0.309 + 0.951i)T \)
11 \( 1 + (3.14 - 1.04i)T \)
13 \( 1 + (0.809 - 0.587i)T \)
good5 \( 1 + (-3.52 - 2.55i)T + (1.54 + 4.75i)T^{2} \)
7 \( 1 + (1.38 - 4.25i)T + (-5.66 - 4.11i)T^{2} \)
17 \( 1 + (-4.08 - 2.96i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (1.58 + 4.87i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + 8.53T + 23T^{2} \)
29 \( 1 + (-0.262 + 0.806i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (0.222 - 0.161i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (1.14 - 3.51i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (2.82 + 8.69i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 8.30T + 43T^{2} \)
47 \( 1 + (2.66 + 8.19i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (7.31 - 5.31i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (1.35 - 4.17i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (5.80 + 4.21i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + 1.93T + 67T^{2} \)
71 \( 1 + (-12.6 - 9.18i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-1.82 + 5.61i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (3.73 - 2.71i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-3.31 - 2.41i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 - 7.68T + 89T^{2} \)
97 \( 1 + (6.27 - 4.55i)T + (29.9 - 92.2i)T^{2} \)
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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

−9.686094511878053125244302239250, −8.949645538858857328937173828877, −7.978318463985642848855719467294, −7.01243989775326854133627840393, −6.20352923243571701132103675362, −5.81087145410569923823109832801, −5.13090543884403650729675465012, −3.26815561855869263595864934676, −2.33513134188065080858820846796, −2.04777630218958109935530084366, 0.49486376533146218548430312061, 1.69893322189185842741844717620, 3.08055321187718998202688579587, 4.19629146460339934916709484569, 4.95984554298572148510214487283, 5.79906844880108730232659412034, 6.31480168661389540175928207113, 7.66535138555748210002680898946, 8.217430220751325955416725536267, 9.443644969928391920375814888057

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