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Properties

Label 2-1716-11.3-c1-0-4
Degree $2$
Conductor $1716$
Sign $0.0890 - 0.996i$
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 + (0.554 + 0.402i)5-s + (−0.256 + 0.788i)7-s + (−0.809 + 0.587i)9-s + (−3.08 − 1.22i)11-s + (−0.809 + 0.587i)13-s + (0.211 − 0.651i)15-s + (0.875 + 0.636i)17-s + (1.65 + 5.10i)19-s + 0.829·21-s + 2.96·23-s + (−1.40 − 4.30i)25-s + (0.809 + 0.587i)27-s + (−0.113 + 0.348i)29-s + (−5.01 + 3.64i)31-s + ⋯
L(s)  = 1  + (−0.178 − 0.549i)3-s + (0.247 + 0.180i)5-s + (−0.0968 + 0.298i)7-s + (−0.269 + 0.195i)9-s + (−0.928 − 0.370i)11-s + (−0.224 + 0.163i)13-s + (0.0546 − 0.168i)15-s + (0.212 + 0.154i)17-s + (0.380 + 1.17i)19-s + 0.180·21-s + 0.617·23-s + (−0.280 − 0.861i)25-s + (0.155 + 0.113i)27-s + (−0.0210 + 0.0646i)29-s + (−0.901 + 0.654i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1716 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0890 - 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1716 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0890 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1716\)    =    \(2^{2} \cdot 3 \cdot 11 \cdot 13\)
Sign: $0.0890 - 0.996i$
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: \((2,\ 1716,\ (\ :1/2),\ 0.0890 - 0.996i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9765393187\)
\(L(\frac12)\) \(\approx\) \(0.9765393187\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.08 + 1.22i)T \)
13 \( 1 + (0.809 - 0.587i)T \)
good5 \( 1 + (-0.554 - 0.402i)T + (1.54 + 4.75i)T^{2} \)
7 \( 1 + (0.256 - 0.788i)T + (-5.66 - 4.11i)T^{2} \)
17 \( 1 + (-0.875 - 0.636i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (-1.65 - 5.10i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 - 2.96T + 23T^{2} \)
29 \( 1 + (0.113 - 0.348i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (5.01 - 3.64i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (2.66 - 8.19i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (-0.590 - 1.81i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + 8.65T + 43T^{2} \)
47 \( 1 + (-3.97 - 12.2i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (-1.77 + 1.29i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (1.26 - 3.89i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (-11.2 - 8.15i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + 10.9T + 67T^{2} \)
71 \( 1 + (10.5 + 7.66i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-3.99 + 12.2i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (0.141 - 0.102i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-11.1 - 8.07i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + 1.85T + 89T^{2} \)
97 \( 1 + (8.93 - 6.49i)T + (29.9 - 92.2i)T^{2} \)
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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

−9.539667958453171149711229167444, −8.584705654838642936192160978943, −7.916138486238202651332845549712, −7.17021833245269425891876928014, −6.21662404137915247379824932714, −5.61557167176914535682113104781, −4.74396644264325585857819370909, −3.38724463020442946197486876417, −2.53033907812988980857270331812, −1.35773996558294653631110613386, 0.37882818308722867218794806395, 2.09745105658949552374762147845, 3.18741453990533724882041378378, 4.16159112088773231102451331576, 5.31131871729391375303432360867, 5.44125468219222593082871593933, 6.95532455285754271205764633025, 7.38324268703239093165769357128, 8.499956642555025770291084990438, 9.261062039728357666514735693651

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