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Properties

Label 2-1716-11.9-c1-0-20
Degree $2$
Conductor $1716$
Sign $-0.773 + 0.633i$
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 + (1.03 + 3.17i)5-s + (−2.71 − 1.97i)7-s + (0.309 − 0.951i)9-s + (−3.16 − 0.985i)11-s + (0.309 − 0.951i)13-s + (2.69 + 1.95i)15-s + (−1.91 − 5.88i)17-s + (−0.0260 + 0.0189i)19-s − 3.35·21-s − 4.94·23-s + (−4.95 + 3.59i)25-s + (−0.309 − 0.951i)27-s + (−3.27 − 2.38i)29-s + (−1.71 + 5.27i)31-s + ⋯
L(s)  = 1  + (0.467 − 0.339i)3-s + (0.460 + 1.41i)5-s + (−1.02 − 0.745i)7-s + (0.103 − 0.317i)9-s + (−0.954 − 0.297i)11-s + (0.0857 − 0.263i)13-s + (0.696 + 0.506i)15-s + (−0.463 − 1.42i)17-s + (−0.00597 + 0.00434i)19-s − 0.732·21-s − 1.03·23-s + (−0.990 + 0.719i)25-s + (−0.0594 − 0.183i)27-s + (−0.608 − 0.442i)29-s + (−0.307 + 0.946i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1716 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.773 + 0.633i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1716 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.773 + 0.633i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1716\)    =    \(2^{2} \cdot 3 \cdot 11 \cdot 13\)
Sign: $-0.773 + 0.633i$
Analytic conductor: \(13.7023\)
Root analytic conductor: \(3.70166\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1716} (625, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1716,\ (\ :1/2),\ -0.773 + 0.633i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6132757703\)
\(L(\frac12)\) \(\approx\) \(0.6132757703\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.16 + 0.985i)T \)
13 \( 1 + (-0.309 + 0.951i)T \)
good5 \( 1 + (-1.03 - 3.17i)T + (-4.04 + 2.93i)T^{2} \)
7 \( 1 + (2.71 + 1.97i)T + (2.16 + 6.65i)T^{2} \)
17 \( 1 + (1.91 + 5.88i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (0.0260 - 0.0189i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + 4.94T + 23T^{2} \)
29 \( 1 + (3.27 + 2.38i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (1.71 - 5.27i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (-7.03 - 5.10i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (-2.60 + 1.89i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + 12.8T + 43T^{2} \)
47 \( 1 + (0.0961 - 0.0698i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (-2.27 + 6.98i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (6.88 + 5.00i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (1.92 + 5.93i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + 3.04T + 67T^{2} \)
71 \( 1 + (4.56 + 14.0i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (1.08 + 0.789i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (-1.88 + 5.79i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (1.11 + 3.43i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 - 7.55T + 89T^{2} \)
97 \( 1 + (3.53 - 10.8i)T + (-78.4 - 57.0i)T^{2} \)
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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

−9.194325015879160011771647316927, −7.974572664925316072905462985982, −7.39054962328815744181042934800, −6.62836988025465881969215113194, −6.15364520936540499284814341249, −4.92358745968495545106156390284, −3.49178374458328633853216916029, −3.03555217929105679710086590075, −2.13455025507293146871752116912, −0.19560995186891806876111444701, 1.71438672885064704890301202416, 2.61256151441103998403505772827, 3.88091822998422749843670957251, 4.63786285685257446490451257985, 5.72204928640013370243110002684, 6.05204294056186122418844839133, 7.44870352057700063407710156887, 8.381926508517003321883928618201, 8.820821365992264212824487505362, 9.615184090031342504698417592718

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