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Properties

Label 2-1710-95.49-c1-0-17
Degree $2$
Conductor $1710$
Sign $-0.827 - 0.561i$
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 + (1.34 + 1.78i)5-s + 4.03i·7-s + 0.999i·8-s + (0.271 + 2.21i)10-s + 1.47·11-s + (−4.38 + 2.53i)13-s + (−2.01 + 3.49i)14-s + (−0.5 + 0.866i)16-s + (0.0812 + 0.0469i)17-s + (−4.00 − 1.72i)19-s + (−0.874 + 2.05i)20-s + (1.27 + 0.735i)22-s + (2.22 − 1.28i)23-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (0.249 + 0.433i)4-s + (0.601 + 0.798i)5-s + 1.52i·7-s + 0.353i·8-s + (0.0859 + 0.701i)10-s + 0.443·11-s + (−1.21 + 0.701i)13-s + (−0.539 + 0.933i)14-s + (−0.125 + 0.216i)16-s + (0.0197 + 0.0113i)17-s + (−0.918 − 0.396i)19-s + (−0.195 + 0.460i)20-s + (0.271 + 0.156i)22-s + (0.463 − 0.267i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1710 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.827 - 0.561i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1710 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.827 - 0.561i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1710\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 19\)
Sign: $-0.827 - 0.561i$
Analytic conductor: \(13.6544\)
Root analytic conductor: \(3.69518\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1710} (1189, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1710,\ (\ :1/2),\ -0.827 - 0.561i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.342689982\)
\(L(\frac12)\) \(\approx\) \(2.342689982\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.34 - 1.78i)T \)
19 \( 1 + (4.00 + 1.72i)T \)
good7 \( 1 - 4.03iT - 7T^{2} \)
11 \( 1 - 1.47T + 11T^{2} \)
13 \( 1 + (4.38 - 2.53i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-0.0812 - 0.0469i)T + (8.5 + 14.7i)T^{2} \)
23 \( 1 + (-2.22 + 1.28i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (2.10 + 3.64i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 4.26T + 31T^{2} \)
37 \( 1 - 1.53iT - 37T^{2} \)
41 \( 1 + (-3.88 + 6.72i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (3.97 + 2.29i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (3.49 - 2.01i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-9.22 + 5.32i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-3.07 + 5.32i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.653 - 1.13i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-9.31 + 5.37i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (4.33 - 7.51i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-11.0 - 6.35i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (6.48 - 11.2i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 2.06iT - 83T^{2} \)
89 \( 1 + (-0.813 - 1.40i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-10.3 - 5.97i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.514711460888239862660284927302, −8.930128343317897421235068451094, −8.024858318698823409108568612722, −6.85909051648928213722853342010, −6.53566068228613778445032926733, −5.58925092179905556458720320183, −4.94296044181027525024747374055, −3.80330855509885954916863780758, −2.48160849991130309659374569882, −2.24229079448463486973194278206, 0.68695512503135179591293763704, 1.76358603438170727910354029113, 3.03629544663829248787062047029, 4.16010895647486488443536557163, 4.70838148664778178707264441103, 5.58314862847841676038731213086, 6.54833762461233433888230217460, 7.32088580147395465389552892759, 8.169222060117443392498076770046, 9.208626199964972640705452798594

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