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Properties

Label 2-1710-19.11-c1-0-3
Degree $2$
Conductor $1710$
Sign $0.403 - 0.914i$
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  + (0.5 − 0.866i)2-s + (−0.499 − 0.866i)4-s + (0.5 − 0.866i)5-s − 4.59·7-s − 0.999·8-s + (−0.499 − 0.866i)10-s − 0.473·11-s + (0.263 + 0.455i)13-s + (−2.29 + 3.98i)14-s + (−0.5 + 0.866i)16-s + (−2.27 + 3.93i)17-s + (2.56 − 3.52i)19-s − 0.999·20-s + (−0.236 + 0.410i)22-s + (−0.236 − 0.410i)23-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.249 − 0.433i)4-s + (0.223 − 0.387i)5-s − 1.73·7-s − 0.353·8-s + (−0.158 − 0.273i)10-s − 0.142·11-s + (0.0730 + 0.126i)13-s + (−0.614 + 1.06i)14-s + (−0.125 + 0.216i)16-s + (−0.551 + 0.954i)17-s + (0.587 − 0.808i)19-s − 0.223·20-s + (−0.0504 + 0.0874i)22-s + (−0.0493 − 0.0855i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1710 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.403 - 0.914i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1710 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.403 - 0.914i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1710\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 19\)
Sign: $0.403 - 0.914i$
Analytic conductor: \(13.6544\)
Root analytic conductor: \(3.69518\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1710} (1531, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1710,\ (\ :1/2),\ 0.403 - 0.914i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6764415603\)
\(L(\frac12)\) \(\approx\) \(0.6764415603\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 + 0.866i)T \)
3 \( 1 \)
5 \( 1 + (-0.5 + 0.866i)T \)
19 \( 1 + (-2.56 + 3.52i)T \)
good7 \( 1 + 4.59T + 7T^{2} \)
11 \( 1 + 0.473T + 11T^{2} \)
13 \( 1 + (-0.263 - 0.455i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (2.27 - 3.93i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (0.236 + 0.410i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-4.32 - 7.49i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 0.526T + 31T^{2} \)
37 \( 1 + T + 37T^{2} \)
41 \( 1 + (5.63 - 9.75i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (6.33 - 10.9i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-4.07 - 7.05i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-0.964 - 1.66i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-3 + 5.19i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (1.53 + 2.66i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (4.53 + 7.85i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (2.27 - 3.93i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-1.46 + 2.53i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-0.736 + 1.27i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 4.10T + 83T^{2} \)
89 \( 1 + (0.964 + 1.66i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (8.79 - 15.2i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.483384838558699048505110320309, −9.034622252276630585078550564415, −8.042090530661626768822194697251, −6.66484577030016766403761520352, −6.40201857460489790739428196621, −5.30213969784561633195628648413, −4.43050879506436436806433867890, −3.36303570407595423943006407148, −2.75022171422650488970917553252, −1.31216249679929664331950819900, 0.23015875231207545739589979128, 2.42193136415198379481636951675, 3.30761891089830727333474717912, 4.05629774560629155614909858357, 5.38793715392485758088766396122, 5.95728008336597280089541399545, 6.90717342451600671145415579139, 7.17334484341148813758889137514, 8.408157591399298354417294850457, 9.162534690391240051861603323339

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