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Properties

Label 2-1716-11.4-c1-0-0
Degree $2$
Conductor $1716$
Sign $-0.913 + 0.405i$
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 + (−0.439 + 0.318i)5-s + (0.972 + 2.99i)7-s + (−0.809 − 0.587i)9-s + (−2.15 − 2.51i)11-s + (−0.809 − 0.587i)13-s + (−0.167 − 0.516i)15-s + (0.343 − 0.249i)17-s + (−1.31 + 4.04i)19-s − 3.14·21-s − 2.27·23-s + (−1.45 + 4.47i)25-s + (0.809 − 0.587i)27-s + (−0.366 − 1.12i)29-s + (−6.77 − 4.92i)31-s + ⋯
L(s)  = 1  + (−0.178 + 0.549i)3-s + (−0.196 + 0.142i)5-s + (0.367 + 1.13i)7-s + (−0.269 − 0.195i)9-s + (−0.650 − 0.759i)11-s + (−0.224 − 0.163i)13-s + (−0.0432 − 0.133i)15-s + (0.0832 − 0.0604i)17-s + (−0.301 + 0.927i)19-s − 0.686·21-s − 0.473·23-s + (−0.290 + 0.895i)25-s + (0.155 − 0.113i)27-s + (−0.0681 − 0.209i)29-s + (−1.21 − 0.884i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1716 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.913 + 0.405i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1716 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.913 + 0.405i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1716\)    =    \(2^{2} \cdot 3 \cdot 11 \cdot 13\)
Sign: $-0.913 + 0.405i$
Analytic conductor: \(13.7023\)
Root analytic conductor: \(3.70166\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1716} (1093, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1716,\ (\ :1/2),\ -0.913 + 0.405i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3320030114\)
\(L(\frac12)\) \(\approx\) \(0.3320030114\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.15 + 2.51i)T \)
13 \( 1 + (0.809 + 0.587i)T \)
good5 \( 1 + (0.439 - 0.318i)T + (1.54 - 4.75i)T^{2} \)
7 \( 1 + (-0.972 - 2.99i)T + (-5.66 + 4.11i)T^{2} \)
17 \( 1 + (-0.343 + 0.249i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (1.31 - 4.04i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + 2.27T + 23T^{2} \)
29 \( 1 + (0.366 + 1.12i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (6.77 + 4.92i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-1.78 - 5.48i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (-1.96 + 6.05i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + 6.42T + 43T^{2} \)
47 \( 1 + (2.71 - 8.37i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (0.668 + 0.485i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (3.32 + 10.2i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (1.19 - 0.868i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + 0.293T + 67T^{2} \)
71 \( 1 + (-8.24 + 5.98i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (3.67 + 11.3i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (6.94 + 5.04i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (9.88 - 7.17i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 - 1.06T + 89T^{2} \)
97 \( 1 + (0.662 + 0.481i)T + (29.9 + 92.2i)T^{2} \)
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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

−9.703459331651029603768536668643, −9.044306498874379799679727084453, −8.163672280595741051068165950552, −7.68383351482799744505179593399, −6.30564741882563467031386224469, −5.64009262009446331860319396409, −5.04736250243143015490427428832, −3.87093189474409363978089477607, −3.00030549147519914863763574435, −1.89525870590725710208734773410, 0.12301986691453628977463585489, 1.51608031206511571420114654119, 2.61464499395244553408324570062, 3.97646437229508156827714103924, 4.69503351470376440530606591568, 5.56720308905942486290996678604, 6.78035029612807671365209171101, 7.23495472400401472189670104943, 7.944993353229610768164618993864, 8.718428177440534820643590305618

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