Properties

Label 2-1710-5.4-c1-0-31
Degree $2$
Conductor $1710$
Sign $0.774 - 0.632i$
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  + i·2-s − 4-s + (1.41 + 1.73i)5-s − 1.03i·7-s i·8-s + (−1.73 + 1.41i)10-s + 3.86·11-s − 1.03i·13-s + 1.03·14-s + 16-s − 7.46i·17-s + 19-s + (−1.41 − 1.73i)20-s + 3.86i·22-s − 1.46i·23-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.5·4-s + (0.632 + 0.774i)5-s − 0.391i·7-s − 0.353i·8-s + (−0.547 + 0.447i)10-s + 1.16·11-s − 0.287i·13-s + 0.276·14-s + 0.250·16-s − 1.81i·17-s + 0.229·19-s + (−0.316 − 0.387i)20-s + 0.823i·22-s − 0.305i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1710 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.774 - 0.632i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1710 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.774 - 0.632i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1710\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 19\)
Sign: $0.774 - 0.632i$
Analytic conductor: \(13.6544\)
Root analytic conductor: \(3.69518\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1710} (1369, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1710,\ (\ :1/2),\ 0.774 - 0.632i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.992881465\)
\(L(\frac12)\) \(\approx\) \(1.992881465\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
3 \( 1 \)
5 \( 1 + (-1.41 - 1.73i)T \)
19 \( 1 - T \)
good7 \( 1 + 1.03iT - 7T^{2} \)
11 \( 1 - 3.86T + 11T^{2} \)
13 \( 1 + 1.03iT - 13T^{2} \)
17 \( 1 + 7.46iT - 17T^{2} \)
23 \( 1 + 1.46iT - 23T^{2} \)
29 \( 1 - 9.52T + 29T^{2} \)
31 \( 1 - 2.92T + 31T^{2} \)
37 \( 1 + 6.69iT - 37T^{2} \)
41 \( 1 + 6.69T + 41T^{2} \)
43 \( 1 - 1.79iT - 43T^{2} \)
47 \( 1 + 9.46iT - 47T^{2} \)
53 \( 1 - 6iT - 53T^{2} \)
59 \( 1 + 12.6T + 59T^{2} \)
61 \( 1 + 2T + 61T^{2} \)
67 \( 1 - 3.58iT - 67T^{2} \)
71 \( 1 - 15.4T + 71T^{2} \)
73 \( 1 - 13.3iT - 73T^{2} \)
79 \( 1 - 4T + 79T^{2} \)
83 \( 1 + 4.39iT - 83T^{2} \)
89 \( 1 - 1.03T + 89T^{2} \)
97 \( 1 - 17.2iT - 97T^{2} \)
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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

−9.401020024019400224958548107202, −8.672605890701921395339003358991, −7.60230843437956142285627278560, −6.87970819332043650386342717377, −6.46506136709095214176852501726, −5.45452060855542491889013354948, −4.60733918705424592940343453681, −3.51890558221933551597862946451, −2.53368172209752490319801618265, −0.940057480850032100986087174772, 1.18942348892091815590722486608, 1.91997699699470899929889148461, 3.22341238503343690654358205928, 4.24976403894787311760930207382, 4.94771777850974583771724464556, 6.07907297265899307305344122702, 6.52034948272875569113125076239, 8.077921370227025993999669159633, 8.613393981711110045295796942671, 9.295259581837191283148781332024

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