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Properties

Label 2-1710-15.2-c1-0-13
Degree $2$
Conductor $1710$
Sign $0.313 - 0.949i$
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  + (−0.707 + 0.707i)2-s − 1.00i·4-s + (1.70 + 1.44i)5-s + (0.414 + 0.414i)7-s + (0.707 + 0.707i)8-s + (−2.22 + 0.185i)10-s + 2.58i·11-s + (2.88 − 2.88i)13-s − 0.585·14-s − 1.00·16-s + (1.36 − 1.36i)17-s − i·19-s + (1.44 − 1.70i)20-s + (−1.82 − 1.82i)22-s + (2.56 + 2.56i)23-s + ⋯
L(s)  = 1  + (−0.499 + 0.499i)2-s − 0.500i·4-s + (0.763 + 0.645i)5-s + (0.156 + 0.156i)7-s + (0.250 + 0.250i)8-s + (−0.704 + 0.0587i)10-s + 0.779i·11-s + (0.801 − 0.801i)13-s − 0.156·14-s − 0.250·16-s + (0.331 − 0.331i)17-s − 0.229i·19-s + (0.322 − 0.381i)20-s + (−0.389 − 0.389i)22-s + (0.534 + 0.534i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1710 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.313 - 0.949i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1710 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.313 - 0.949i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1710\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 19\)
Sign: $0.313 - 0.949i$
Analytic conductor: \(13.6544\)
Root analytic conductor: \(3.69518\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1710} (647, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1710,\ (\ :1/2),\ 0.313 - 0.949i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.653379985\)
\(L(\frac12)\) \(\approx\) \(1.653379985\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.707 - 0.707i)T \)
3 \( 1 \)
5 \( 1 + (-1.70 - 1.44i)T \)
19 \( 1 + iT \)
good7 \( 1 + (-0.414 - 0.414i)T + 7iT^{2} \)
11 \( 1 - 2.58iT - 11T^{2} \)
13 \( 1 + (-2.88 + 2.88i)T - 13iT^{2} \)
17 \( 1 + (-1.36 + 1.36i)T - 17iT^{2} \)
23 \( 1 + (-2.56 - 2.56i)T + 23iT^{2} \)
29 \( 1 - 5.77T + 29T^{2} \)
31 \( 1 - 0.645T + 31T^{2} \)
37 \( 1 + (1.41 + 1.41i)T + 37iT^{2} \)
41 \( 1 + 0.302iT - 41T^{2} \)
43 \( 1 + (3.30 - 3.30i)T - 43iT^{2} \)
47 \( 1 + (0.788 - 0.788i)T - 47iT^{2} \)
53 \( 1 + (-4.24 - 4.24i)T + 53iT^{2} \)
59 \( 1 - 5.47T + 59T^{2} \)
61 \( 1 + 6.91T + 61T^{2} \)
67 \( 1 + (1.47 + 1.47i)T + 67iT^{2} \)
71 \( 1 + 0.913iT - 71T^{2} \)
73 \( 1 + (1.82 - 1.82i)T - 73iT^{2} \)
79 \( 1 + 0.217iT - 79T^{2} \)
83 \( 1 + (-9.69 - 9.69i)T + 83iT^{2} \)
89 \( 1 - 5.04T + 89T^{2} \)
97 \( 1 + (-9.40 - 9.40i)T + 97iT^{2} \)
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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

−9.491770149195716624084646252585, −8.725304349496497145635151056899, −7.86084834543052094643946110491, −7.08588139654730034759830646564, −6.39201436674885289023830570684, −5.57313210769709776970915724295, −4.84363373500420433701350456264, −3.41917921856078413381525068503, −2.38993099853000422628490843111, −1.19320232988231117018612968722, 0.897722616859884329363770508752, 1.81484521690628339797421659065, 3.01899821511613317003086743304, 4.08039062430342327180622564434, 5.00330029996740569328017975124, 6.03993102439581400468012749213, 6.68871702189790092189428949975, 7.898029677432119882080235868856, 8.694537613723569791961395766863, 8.984092216650935650819900807706

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