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Properties

Label 2-1710-95.64-c1-0-48
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: $\chi_{1710} (919, \cdot )$
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

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

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