Properties

Label 2-1710-19.7-c1-0-17
Degree $2$
Conductor $1710$
Sign $-0.0977 + 0.995i$
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 − 5·7-s + 0.999·8-s + (0.499 − 0.866i)10-s − 11-s + (−3 + 5.19i)13-s + (2.5 + 4.33i)14-s + (−0.5 − 0.866i)16-s + (2 + 3.46i)17-s + (−0.5 − 4.33i)19-s − 0.999·20-s + (0.5 + 0.866i)22-s + (3.5 − 6.06i)23-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.249 + 0.433i)4-s + (0.223 + 0.387i)5-s − 1.88·7-s + 0.353·8-s + (0.158 − 0.273i)10-s − 0.301·11-s + (−0.832 + 1.44i)13-s + (0.668 + 1.15i)14-s + (−0.125 − 0.216i)16-s + (0.485 + 0.840i)17-s + (−0.114 − 0.993i)19-s − 0.223·20-s + (0.106 + 0.184i)22-s + (0.729 − 1.26i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1710 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0977 + 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1710 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0977 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1710\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 19\)
Sign: $-0.0977 + 0.995i$
Analytic conductor: \(13.6544\)
Root analytic conductor: \(3.69518\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1710} (1261, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1710,\ (\ :1/2),\ -0.0977 + 0.995i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6825048764\)
\(L(\frac12)\) \(\approx\) \(0.6825048764\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.5 + 4.33i)T \)
good7 \( 1 + 5T + 7T^{2} \)
11 \( 1 + T + 11T^{2} \)
13 \( 1 + (3 - 5.19i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-2 - 3.46i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (-3.5 + 6.06i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3 + 5.19i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 - 7T + 37T^{2} \)
41 \( 1 + (2.5 + 4.33i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (3 + 5.19i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (4 - 6.92i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-5.5 + 9.52i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (4 + 6.92i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2 + 3.46i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (6 - 10.3i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-1 - 1.73i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-1 - 1.73i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (5 + 8.66i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 10T + 83T^{2} \)
89 \( 1 + (-6.5 + 11.2i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (1 + 1.73i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.266738264866538870634634773207, −8.674753056178696405250714619088, −7.43366043197413088061239649776, −6.67138538879137382532202686114, −6.21472855405825764037049207558, −4.81316911907771002250255011138, −3.85939317970140680894921823197, −2.89382931247378853677318609190, −2.22309201653286083952235233638, −0.36994564203229181857154935077, 0.921348307351019644856337763920, 2.79838686637030373939818071606, 3.44533408316776937831906264339, 4.92603064421680306845631958872, 5.62887984335819565507149161373, 6.30450335167263285415955961060, 7.26672082479702954143164479571, 7.79775482413170400776768667274, 8.844054411717622077399636632895, 9.662880775712459049022105377144

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