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Properties

Label 2-1710-95.64-c1-0-11
Degree $2$
Conductor $1710$
Sign $0.920 - 0.390i$
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  + (0.866 − 0.5i)2-s + (0.499 − 0.866i)4-s + (−1.99 − 1.01i)5-s + 2.79i·7-s − 0.999i·8-s + (−2.23 + 0.118i)10-s − 4.02·11-s + (−0.0960 − 0.0554i)13-s + (1.39 + 2.42i)14-s + (−0.5 − 0.866i)16-s + (3.68 − 2.12i)17-s + (0.163 + 4.35i)19-s + (−1.87 + 1.21i)20-s + (−3.48 + 2.01i)22-s + (7.65 + 4.42i)23-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (0.249 − 0.433i)4-s + (−0.891 − 0.453i)5-s + 1.05i·7-s − 0.353i·8-s + (−0.706 + 0.0375i)10-s − 1.21·11-s + (−0.0266 − 0.0153i)13-s + (0.373 + 0.647i)14-s + (−0.125 − 0.216i)16-s + (0.893 − 0.515i)17-s + (0.0376 + 0.999i)19-s + (−0.419 + 0.272i)20-s + (−0.743 + 0.429i)22-s + (1.59 + 0.921i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1710 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.920 - 0.390i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1710 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.920 - 0.390i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1710\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 19\)
Sign: $0.920 - 0.390i$
Analytic conductor: \(13.6544\)
Root analytic conductor: \(3.69518\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1710} (919, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1710,\ (\ :1/2),\ 0.920 - 0.390i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.806420476\)
\(L(\frac12)\) \(\approx\) \(1.806420476\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 + 0.5i)T \)
3 \( 1 \)
5 \( 1 + (1.99 + 1.01i)T \)
19 \( 1 + (-0.163 - 4.35i)T \)
good7 \( 1 - 2.79iT - 7T^{2} \)
11 \( 1 + 4.02T + 11T^{2} \)
13 \( 1 + (0.0960 + 0.0554i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-3.68 + 2.12i)T + (8.5 - 14.7i)T^{2} \)
23 \( 1 + (-7.65 - 4.42i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.907 - 1.57i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 7.88T + 31T^{2} \)
37 \( 1 - 1.68iT - 37T^{2} \)
41 \( 1 + (-3.88 - 6.72i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-6.46 + 3.73i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (4.17 + 2.40i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (6.45 + 3.72i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-0.188 - 0.326i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (6.18 - 10.7i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.82 - 2.21i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-5.90 - 10.2i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (8.33 - 4.81i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (2.11 + 3.66i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 8.25iT - 83T^{2} \)
89 \( 1 + (5.01 - 8.68i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-4.52 + 2.61i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.443027320430021376669659082587, −8.510326271537858702211881392456, −7.85104174445569048830693439603, −7.06795199029086942980230014125, −5.74807144128292920049276851486, −5.29126490329726296264644556361, −4.49224458177349418803166785296, −3.28315457845683432216654532775, −2.69187967394828610322135757728, −1.17027029811895879288272655661, 0.64410086177251526382759417142, 2.67304165738464351959451647629, 3.35380935896410874986084615339, 4.42686599781963004947963355815, 4.93071895385025973926184075480, 6.16724146213001928570878822863, 6.97050032221838649742840882659, 7.65008475248054126553669180816, 8.069566604161771011212314177084, 9.177568536841598483991811056293

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