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Properties

Label 2-1716-13.5-c2-0-26
Degree $2$
Conductor $1716$
Sign $0.889 + 0.457i$
Analytic cond. $46.7576$
Root an. cond. $6.83795$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.73·3-s + (−4.83 + 4.83i)5-s + (7.59 + 7.59i)7-s + 2.99·9-s + (−2.34 − 2.34i)11-s + (−0.217 − 12.9i)13-s + (8.38 − 8.38i)15-s + 17.7i·17-s + (22.9 − 22.9i)19-s + (−13.1 − 13.1i)21-s − 40.9i·23-s − 21.8i·25-s − 5.19·27-s − 55.3·29-s + (−25.5 + 25.5i)31-s + ⋯
L(s)  = 1  − 0.577·3-s + (−0.967 + 0.967i)5-s + (1.08 + 1.08i)7-s + 0.333·9-s + (−0.213 − 0.213i)11-s + (−0.0167 − 0.999i)13-s + (0.558 − 0.558i)15-s + 1.04i·17-s + (1.20 − 1.20i)19-s + (−0.626 − 0.626i)21-s − 1.78i·23-s − 0.873i·25-s − 0.192·27-s − 1.91·29-s + (−0.822 + 0.822i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1716 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.889 + 0.457i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1716 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.889 + 0.457i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1716\)    =    \(2^{2} \cdot 3 \cdot 11 \cdot 13\)
Sign: $0.889 + 0.457i$
Analytic conductor: \(46.7576\)
Root analytic conductor: \(6.83795\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1716} (265, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1716,\ (\ :1),\ 0.889 + 0.457i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.140067832\)
\(L(\frac12)\) \(\approx\) \(1.140067832\)
\(L(2)\) 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 \( 1 + 1.73T \)
11 \( 1 + (2.34 + 2.34i)T \)
13 \( 1 + (0.217 + 12.9i)T \)
good5 \( 1 + (4.83 - 4.83i)T - 25iT^{2} \)
7 \( 1 + (-7.59 - 7.59i)T + 49iT^{2} \)
17 \( 1 - 17.7iT - 289T^{2} \)
19 \( 1 + (-22.9 + 22.9i)T - 361iT^{2} \)
23 \( 1 + 40.9iT - 529T^{2} \)
29 \( 1 + 55.3T + 841T^{2} \)
31 \( 1 + (25.5 - 25.5i)T - 961iT^{2} \)
37 \( 1 + (-12.3 - 12.3i)T + 1.36e3iT^{2} \)
41 \( 1 + (-17.9 + 17.9i)T - 1.68e3iT^{2} \)
43 \( 1 + 45.1iT - 1.84e3T^{2} \)
47 \( 1 + (60.1 + 60.1i)T + 2.20e3iT^{2} \)
53 \( 1 - 105.T + 2.80e3T^{2} \)
59 \( 1 + (-19.4 - 19.4i)T + 3.48e3iT^{2} \)
61 \( 1 - 73.8T + 3.72e3T^{2} \)
67 \( 1 + (-55.0 + 55.0i)T - 4.48e3iT^{2} \)
71 \( 1 + (57.1 - 57.1i)T - 5.04e3iT^{2} \)
73 \( 1 + (36.3 + 36.3i)T + 5.32e3iT^{2} \)
79 \( 1 - 56.7T + 6.24e3T^{2} \)
83 \( 1 + (12.8 - 12.8i)T - 6.88e3iT^{2} \)
89 \( 1 + (86.3 + 86.3i)T + 7.92e3iT^{2} \)
97 \( 1 + (68.5 - 68.5i)T - 9.40e3iT^{2} \)
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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

−8.820316699254457794704835702868, −8.298914254949407247882847860963, −7.43702541984974685860384069444, −6.83266055279354067427872586686, −5.59921995469394386755967783230, −5.24412026892636381542939520790, −4.04109008560870829782033221960, −3.07712492153065290218958584675, −2.06650301036739839456601340754, −0.42895794202631746362704739873, 0.910251339836120238696033566275, 1.69548749812041942650170186924, 3.70853444294893923498295242970, 4.21269379870608959374679714910, 5.06516287326145607331762431232, 5.67417177357401620367129238497, 7.29924728782447935305234642276, 7.46644161006821513968780218933, 8.151681881772871856179537017226, 9.400392858434837059356245183851

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