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Properties

Label 2-1716-11.5-c1-0-13
Degree $2$
Conductor $1716$
Sign $0.184 + 0.982i$
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  + (−0.809 − 0.587i)3-s + (−0.558 + 1.71i)5-s + (−2.68 + 1.95i)7-s + (0.309 + 0.951i)9-s + (−3.05 − 1.28i)11-s + (0.309 + 0.951i)13-s + (1.46 − 1.06i)15-s + (−0.849 + 2.61i)17-s + (−5.23 − 3.80i)19-s + 3.32·21-s + 0.621·23-s + (1.40 + 1.01i)25-s + (0.309 − 0.951i)27-s + (5.95 − 4.32i)29-s + (−0.498 − 1.53i)31-s + ⋯
L(s)  = 1  + (−0.467 − 0.339i)3-s + (−0.249 + 0.768i)5-s + (−1.01 + 0.738i)7-s + (0.103 + 0.317i)9-s + (−0.922 − 0.386i)11-s + (0.0857 + 0.263i)13-s + (0.377 − 0.274i)15-s + (−0.205 + 0.633i)17-s + (−1.20 − 0.872i)19-s + 0.725·21-s + 0.129·23-s + (0.280 + 0.203i)25-s + (0.0594 − 0.183i)27-s + (1.10 − 0.803i)29-s + (−0.0895 − 0.275i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1716 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.184 + 0.982i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1716 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.184 + 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1716\)    =    \(2^{2} \cdot 3 \cdot 11 \cdot 13\)
Sign: $0.184 + 0.982i$
Analytic conductor: \(13.7023\)
Root analytic conductor: \(3.70166\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1716} (313, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1716,\ (\ :1/2),\ 0.184 + 0.982i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5390787986\)
\(L(\frac12)\) \(\approx\) \(0.5390787986\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.809 + 0.587i)T \)
11 \( 1 + (3.05 + 1.28i)T \)
13 \( 1 + (-0.309 - 0.951i)T \)
good5 \( 1 + (0.558 - 1.71i)T + (-4.04 - 2.93i)T^{2} \)
7 \( 1 + (2.68 - 1.95i)T + (2.16 - 6.65i)T^{2} \)
17 \( 1 + (0.849 - 2.61i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (5.23 + 3.80i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 - 0.621T + 23T^{2} \)
29 \( 1 + (-5.95 + 4.32i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (0.498 + 1.53i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (-6.24 + 4.53i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (6.59 + 4.79i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 - 3.09T + 43T^{2} \)
47 \( 1 + (0.130 + 0.0946i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (-2.36 - 7.29i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (-0.303 + 0.220i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (-3.42 + 10.5i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 - 6.97T + 67T^{2} \)
71 \( 1 + (0.348 - 1.07i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (-4.17 + 3.03i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (0.406 + 1.25i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (-3.61 + 11.1i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 + 5.66T + 89T^{2} \)
97 \( 1 + (2.60 + 8.01i)T + (-78.4 + 57.0i)T^{2} \)
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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

−9.098855937333245044193628779407, −8.368750290467186807886871869173, −7.45304565343147950750603278099, −6.52338804901277722313619803534, −6.20902491286591621361321837128, −5.21714569273386726173874728732, −4.08584804461612689056246370215, −2.92999871623983514542342386470, −2.28243989804650648593275032859, −0.26896632685202343939712146895, 0.941694934799440504494648373279, 2.66872472262609066385018638611, 3.75357931974312717367884213175, 4.61134757553207836605799869247, 5.26358090791126062900277644712, 6.38150385175371438789982626291, 6.95676970420029202153051912819, 8.044384787514169913257877842195, 8.660430917813069584339270029180, 9.725476487758622311479706322179

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