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Properties

Label 2-1710-19.11-c1-0-0
Degree $2$
Conductor $1710$
Sign $-0.942 + 0.334i$
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 + 2.64·7-s + 0.999·8-s + (−0.499 − 0.866i)10-s − 4.29·11-s + (−1.32 + 2.29i)14-s + (−0.5 + 0.866i)16-s + (−0.177 + 0.306i)17-s + (−4.32 + 0.559i)19-s + 0.999·20-s + (2.14 − 3.71i)22-s + (0.5 + 0.866i)23-s + (−0.499 − 0.866i)25-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.249 − 0.433i)4-s + (−0.223 + 0.387i)5-s + 0.999·7-s + 0.353·8-s + (−0.158 − 0.273i)10-s − 1.29·11-s + (−0.353 + 0.612i)14-s + (−0.125 + 0.216i)16-s + (−0.0429 + 0.0744i)17-s + (−0.991 + 0.128i)19-s + 0.223·20-s + (0.457 − 0.792i)22-s + (0.104 + 0.180i)23-s + (−0.0999 − 0.173i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1710 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.942 + 0.334i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1710 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.942 + 0.334i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1710\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 19\)
Sign: $-0.942 + 0.334i$
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: \((2,\ 1710,\ (\ :1/2),\ -0.942 + 0.334i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3509361576\)
\(L(\frac12)\) \(\approx\) \(0.3509361576\)
\(L(\frac{3}{2})\) 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 \( 1 + (4.32 - 0.559i)T \)
good7 \( 1 - 2.64T + 7T^{2} \)
11 \( 1 + 4.29T + 11T^{2} \)
13 \( 1 + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (0.177 - 0.306i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (-0.5 - 0.866i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.822 - 1.42i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 3.29T + 31T^{2} \)
37 \( 1 + 2.29T + 37T^{2} \)
41 \( 1 + (1.67 - 2.90i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (3 - 5.19i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (4.29 + 7.43i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-0.322 - 0.559i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (2.64 - 4.58i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4.46 - 7.73i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.17 - 3.77i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (4.17 - 7.23i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (4.17 - 7.23i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-2.64 + 4.58i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 4.58T + 83T^{2} \)
89 \( 1 + (6.61 + 11.4i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (1.82 - 3.15i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.810633377143034201271075600776, −8.603568554420389963637278495226, −8.238303829236813866223805780977, −7.46528366120837376441051397722, −6.75976587265981493941436200551, −5.70078584895146846238500801338, −5.01874401363501227110471462294, −4.13954082151218415006788704687, −2.79778455346513319384314623791, −1.64479483142833181694011783356, 0.14395639064320048591418109610, 1.66263350875851952994921656541, 2.56751355867798412765949477418, 3.77657042373879152696206428466, 4.78558750643055533607138388695, 5.28704370373019647863179222652, 6.59173510474521832246883560980, 7.72564061068625973391838958104, 8.122033598651573226469232338407, 8.835421564885575988439233675755

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