Properties

Label 2-1710-5.4-c1-0-27
Degree $2$
Conductor $1710$
Sign $0.662 - 0.749i$
Analytic cond. $13.6544$
Root an. cond. $3.69518$
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  + i·2-s − 4-s + (1.67 + 1.48i)5-s − 3.35i·7-s i·8-s + (−1.48 + 1.67i)10-s + 0.962·11-s + 1.61i·13-s + 3.35·14-s + 16-s + 0.387i·17-s − 19-s + (−1.67 − 1.48i)20-s + 0.962i·22-s − 0.962i·23-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.5·4-s + (0.749 + 0.662i)5-s − 1.26i·7-s − 0.353i·8-s + (−0.468 + 0.529i)10-s + 0.290·11-s + 0.447i·13-s + 0.895·14-s + 0.250·16-s + 0.0940i·17-s − 0.229·19-s + (−0.374 − 0.331i)20-s + 0.205i·22-s − 0.200i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1710 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.662 - 0.749i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1710 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.662 - 0.749i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1710\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 19\)
Sign: $0.662 - 0.749i$
Analytic conductor: \(13.6544\)
Root analytic conductor: \(3.69518\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1710} (1369, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1710,\ (\ :1/2),\ 0.662 - 0.749i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.930082513\)
\(L(\frac12)\) \(\approx\) \(1.930082513\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
3 \( 1 \)
5 \( 1 + (-1.67 - 1.48i)T \)
19 \( 1 + T \)
good7 \( 1 + 3.35iT - 7T^{2} \)
11 \( 1 - 0.962T + 11T^{2} \)
13 \( 1 - 1.61iT - 13T^{2} \)
17 \( 1 - 0.387iT - 17T^{2} \)
23 \( 1 + 0.962iT - 23T^{2} \)
29 \( 1 - 6.96T + 29T^{2} \)
31 \( 1 - 3.35T + 31T^{2} \)
37 \( 1 + 1.61iT - 37T^{2} \)
41 \( 1 - 9.27T + 41T^{2} \)
43 \( 1 + 6.18iT - 43T^{2} \)
47 \( 1 + 0.962iT - 47T^{2} \)
53 \( 1 - 6iT - 53T^{2} \)
59 \( 1 - 10.3T + 59T^{2} \)
61 \( 1 - 11.9T + 61T^{2} \)
67 \( 1 + 7.22iT - 67T^{2} \)
71 \( 1 + 7.22T + 71T^{2} \)
73 \( 1 - 3.22iT - 73T^{2} \)
79 \( 1 - 3.35T + 79T^{2} \)
83 \( 1 - 15.0iT - 83T^{2} \)
89 \( 1 - 4.64T + 89T^{2} \)
97 \( 1 + 10.9iT - 97T^{2} \)
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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

−9.451383012933155287369233791955, −8.597167541124059004999277480428, −7.65844727573707313166878999163, −6.89227924653223515635763654787, −6.48280300762636230151135152590, −5.53644324502021762049461089119, −4.46410380432713379481527171174, −3.73718108759838023813831609787, −2.46178742754807222267640820671, −1.00399891866353753784450633822, 1.01105291670493219602777836254, 2.21204272687266732311143998992, 2.90840349330703420590720401256, 4.26586968098719011463317028649, 5.13683674287577788071854459083, 5.80023459051981853608501175446, 6.59277484829343264116401079694, 8.048595097331937704988593062143, 8.620451228864850096933132115179, 9.294366095063741198714805488092

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