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Properties

Label 2-1710-95.18-c1-0-24
Degree $2$
Conductor $1710$
Sign $0.770 + 0.637i$
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 + (−0.114 − 2.23i)5-s + (−1.40 + 1.40i)7-s + (0.707 − 0.707i)8-s + (−1.49 + 1.66i)10-s + 5.29·11-s + (1.91 − 1.91i)13-s + 1.98·14-s − 1.00·16-s + (−0.488 + 0.488i)17-s + (2.44 + 3.60i)19-s + (2.23 − 0.114i)20-s + (−3.74 − 3.74i)22-s + (3.88 + 3.88i)23-s + ⋯
L(s)  = 1  + (−0.499 − 0.499i)2-s + 0.500i·4-s + (−0.0512 − 0.998i)5-s + (−0.530 + 0.530i)7-s + (0.250 − 0.250i)8-s + (−0.473 + 0.524i)10-s + 1.59·11-s + (0.532 − 0.532i)13-s + 0.530·14-s − 0.250·16-s + (−0.118 + 0.118i)17-s + (0.561 + 0.827i)19-s + (0.499 − 0.0256i)20-s + (−0.797 − 0.797i)22-s + (0.810 + 0.810i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1710 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.770 + 0.637i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1710 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.770 + 0.637i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1710\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 19\)
Sign: $0.770 + 0.637i$
Analytic conductor: \(13.6544\)
Root analytic conductor: \(3.69518\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1710} (1063, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1710,\ (\ :1/2),\ 0.770 + 0.637i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.370750256\)
\(L(\frac12)\) \(\approx\) \(1.370750256\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.114 + 2.23i)T \)
19 \( 1 + (-2.44 - 3.60i)T \)
good7 \( 1 + (1.40 - 1.40i)T - 7iT^{2} \)
11 \( 1 - 5.29T + 11T^{2} \)
13 \( 1 + (-1.91 + 1.91i)T - 13iT^{2} \)
17 \( 1 + (0.488 - 0.488i)T - 17iT^{2} \)
23 \( 1 + (-3.88 - 3.88i)T + 23iT^{2} \)
29 \( 1 + 6.19T + 29T^{2} \)
31 \( 1 - 7.84iT - 31T^{2} \)
37 \( 1 + (5.60 + 5.60i)T + 37iT^{2} \)
41 \( 1 - 1.90iT - 41T^{2} \)
43 \( 1 + (-7.12 - 7.12i)T + 43iT^{2} \)
47 \( 1 + (-7.86 + 7.86i)T - 47iT^{2} \)
53 \( 1 + (-8.93 + 8.93i)T - 53iT^{2} \)
59 \( 1 + 0.611T + 59T^{2} \)
61 \( 1 - 8.31T + 61T^{2} \)
67 \( 1 + (10.2 + 10.2i)T + 67iT^{2} \)
71 \( 1 + 3.97iT - 71T^{2} \)
73 \( 1 + (-7.72 - 7.72i)T + 73iT^{2} \)
79 \( 1 - 17.1T + 79T^{2} \)
83 \( 1 + (5.43 + 5.43i)T + 83iT^{2} \)
89 \( 1 - 5.42T + 89T^{2} \)
97 \( 1 + (-3.10 - 3.10i)T + 97iT^{2} \)
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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

−9.065731278673651783599548690185, −8.885765560961432212599566180403, −7.86373162593241672596763683690, −6.96171448552616724857178119428, −5.94290748937021900369376672026, −5.21395887666010902532501849278, −3.89856158597610181115002410949, −3.40503265117544321126524510739, −1.84307664942865687205695623814, −0.946322627973314659724928125183, 0.881091391610153892628912436292, 2.34189809038708376585198455523, 3.61110427333323930830687512258, 4.26288802426513629549527443658, 5.70151249833535650326269840427, 6.51094681700884739864181958217, 6.98565023084001890425148740578, 7.55519451028824263358316853015, 8.914305219782945951294064567695, 9.202867187907881802401491787789

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