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Properties

Label 2-1710-95.18-c1-0-35
Degree $2$
Conductor $1710$
Sign $0.859 - 0.510i$
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 + (−1.25 + 1.84i)5-s + (3.10 − 3.10i)7-s + (−0.707 + 0.707i)8-s + (−2.19 + 0.416i)10-s + 3.82·11-s + (0.0891 − 0.0891i)13-s + 4.39·14-s − 1.00·16-s + (1.83 − 1.83i)17-s + (2.70 − 3.41i)19-s + (−1.84 − 1.25i)20-s + (2.70 + 2.70i)22-s + (−2.58 − 2.58i)23-s + ⋯
L(s)  = 1  + (0.499 + 0.499i)2-s + 0.500i·4-s + (−0.562 + 0.826i)5-s + (1.17 − 1.17i)7-s + (−0.250 + 0.250i)8-s + (−0.694 + 0.131i)10-s + 1.15·11-s + (0.0247 − 0.0247i)13-s + 1.17·14-s − 0.250·16-s + (0.443 − 0.443i)17-s + (0.620 − 0.784i)19-s + (−0.413 − 0.281i)20-s + (0.576 + 0.576i)22-s + (−0.539 − 0.539i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1710 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.859 - 0.510i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1710 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.859 - 0.510i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1710\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 19\)
Sign: $0.859 - 0.510i$
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: \((2,\ 1710,\ (\ :1/2),\ 0.859 - 0.510i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.529235269\)
\(L(\frac12)\) \(\approx\) \(2.529235269\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.25 - 1.84i)T \)
19 \( 1 + (-2.70 + 3.41i)T \)
good7 \( 1 + (-3.10 + 3.10i)T - 7iT^{2} \)
11 \( 1 - 3.82T + 11T^{2} \)
13 \( 1 + (-0.0891 + 0.0891i)T - 13iT^{2} \)
17 \( 1 + (-1.83 + 1.83i)T - 17iT^{2} \)
23 \( 1 + (2.58 + 2.58i)T + 23iT^{2} \)
29 \( 1 - 3.60T + 29T^{2} \)
31 \( 1 + 3.60iT - 31T^{2} \)
37 \( 1 + (7.07 + 7.07i)T + 37iT^{2} \)
41 \( 1 - 11.2iT - 41T^{2} \)
43 \( 1 + (-7.93 - 7.93i)T + 43iT^{2} \)
47 \( 1 + (-0.463 + 0.463i)T - 47iT^{2} \)
53 \( 1 + (3.21 - 3.21i)T - 53iT^{2} \)
59 \( 1 - 9.40T + 59T^{2} \)
61 \( 1 - 8.21T + 61T^{2} \)
67 \( 1 + (-8.78 - 8.78i)T + 67iT^{2} \)
71 \( 1 - 1.66iT - 71T^{2} \)
73 \( 1 + (3.64 + 3.64i)T + 73iT^{2} \)
79 \( 1 - 8.82T + 79T^{2} \)
83 \( 1 + (-0.347 - 0.347i)T + 83iT^{2} \)
89 \( 1 + 9.79T + 89T^{2} \)
97 \( 1 + (-6.88 - 6.88i)T + 97iT^{2} \)
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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

−9.354653539072294181903940900991, −8.237398314641261032258773315616, −7.69231147999388220732066972391, −7.00246996092058648238001240493, −6.42314818994899698366168355261, −5.20375037905457045237224067800, −4.27774727381306488390471463685, −3.81950741249633826752803100397, −2.62158517421680889350610386714, −1.02115089396666829268939922760, 1.24371159933570713410706137037, 2.00534270912952353013363307360, 3.50500854291196442945950411020, 4.17963952331369974288445300540, 5.29011812319012261598565287731, 5.53465339515326694080320918077, 6.78861998075213257001536969170, 7.932737167635658219114793100700, 8.559349238992586075742253137455, 9.132796117398357832622150139317

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