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Properties

Label 2-1710-95.37-c1-0-10
Degree $2$
Conductor $1710$
Sign $0.462 - 0.886i$
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 + (1.66 − 1.49i)5-s + (−0.170 − 0.170i)7-s + (0.707 + 0.707i)8-s + (−0.120 + 2.23i)10-s − 3.43·11-s + (4.54 + 4.54i)13-s + 0.240·14-s − 1.00·16-s + (−0.537 − 0.537i)17-s + (2.42 + 3.62i)19-s + (−1.49 − 1.66i)20-s + (2.42 − 2.42i)22-s + (−5.15 + 5.15i)23-s + ⋯
L(s)  = 1  + (−0.499 + 0.499i)2-s − 0.500i·4-s + (0.744 − 0.668i)5-s + (−0.0643 − 0.0643i)7-s + (0.250 + 0.250i)8-s + (−0.0380 + 0.706i)10-s − 1.03·11-s + (1.25 + 1.25i)13-s + 0.0643·14-s − 0.250·16-s + (−0.130 − 0.130i)17-s + (0.556 + 0.830i)19-s + (−0.334 − 0.372i)20-s + (0.517 − 0.517i)22-s + (−1.07 + 1.07i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1710 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.462 - 0.886i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1710 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.462 - 0.886i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1710\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 19\)
Sign: $0.462 - 0.886i$
Analytic conductor: \(13.6544\)
Root analytic conductor: \(3.69518\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1710} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1710,\ (\ :1/2),\ 0.462 - 0.886i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.393130563\)
\(L(\frac12)\) \(\approx\) \(1.393130563\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.66 + 1.49i)T \)
19 \( 1 + (-2.42 - 3.62i)T \)
good7 \( 1 + (0.170 + 0.170i)T + 7iT^{2} \)
11 \( 1 + 3.43T + 11T^{2} \)
13 \( 1 + (-4.54 - 4.54i)T + 13iT^{2} \)
17 \( 1 + (0.537 + 0.537i)T + 17iT^{2} \)
23 \( 1 + (5.15 - 5.15i)T - 23iT^{2} \)
29 \( 1 - 5.37T + 29T^{2} \)
31 \( 1 - 5.37iT - 31T^{2} \)
37 \( 1 + (5.54 - 5.54i)T - 37iT^{2} \)
41 \( 1 + 3.68iT - 41T^{2} \)
43 \( 1 + (-2.29 + 2.29i)T - 43iT^{2} \)
47 \( 1 + (-9.57 - 9.57i)T + 47iT^{2} \)
53 \( 1 + (-1.93 - 1.93i)T + 53iT^{2} \)
59 \( 1 - 4.20T + 59T^{2} \)
61 \( 1 - 1.65T + 61T^{2} \)
67 \( 1 + (-0.481 + 0.481i)T - 67iT^{2} \)
71 \( 1 - 8.93iT - 71T^{2} \)
73 \( 1 + (-8.74 + 8.74i)T - 73iT^{2} \)
79 \( 1 - 1.15T + 79T^{2} \)
83 \( 1 + (-9.97 + 9.97i)T - 83iT^{2} \)
89 \( 1 + 2.10T + 89T^{2} \)
97 \( 1 + (-10.8 + 10.8i)T - 97iT^{2} \)
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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

−9.336229537858985788865403927503, −8.670778639785826314035385820800, −8.063262475056887108018169265874, −7.09466611185242472011703479159, −6.19686381382453201604714724607, −5.59271707509867225092668025548, −4.74411163848969219463337811106, −3.63362273902904771627688465524, −2.12385495495238815501224928426, −1.18891436115489559784152402604, 0.70277546661185463656287733277, 2.25070361136911718592159766402, 2.88288555717907046750875649287, 3.89684371407431079851532533323, 5.25222632085809129110556121810, 5.97444269643236993331181794150, 6.83173175691019052612821934195, 7.83347758051250537538592422993, 8.409609771839241599568604684126, 9.304296286755974188787977988291

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