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Properties

Label 2-1710-95.37-c1-0-7
Degree $2$
Conductor $1710$
Sign $-0.0106 - 0.999i$
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 + (−1.42 + 1.72i)5-s + (3.40 + 3.40i)7-s + (−0.707 − 0.707i)8-s + (0.215 + 2.22i)10-s − 3.94·11-s + (−4.03 − 4.03i)13-s + 4.81·14-s − 1.00·16-s + (3.90 + 3.90i)17-s + (3.49 + 2.60i)19-s + (1.72 + 1.42i)20-s + (−2.78 + 2.78i)22-s + (−0.537 + 0.537i)23-s + ⋯
L(s)  = 1  + (0.499 − 0.499i)2-s − 0.500i·4-s + (−0.635 + 0.772i)5-s + (1.28 + 1.28i)7-s + (−0.250 − 0.250i)8-s + (0.0682 + 0.703i)10-s − 1.18·11-s + (−1.11 − 1.11i)13-s + 1.28·14-s − 0.250·16-s + (0.947 + 0.947i)17-s + (0.802 + 0.596i)19-s + (0.386 + 0.317i)20-s + (−0.594 + 0.594i)22-s + (−0.111 + 0.111i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1710 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0106 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1710 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0106 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1710\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 19\)
Sign: $-0.0106 - 0.999i$
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: \((2,\ 1710,\ (\ :1/2),\ -0.0106 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.434836432\)
\(L(\frac12)\) \(\approx\) \(1.434836432\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.42 - 1.72i)T \)
19 \( 1 + (-3.49 - 2.60i)T \)
good7 \( 1 + (-3.40 - 3.40i)T + 7iT^{2} \)
11 \( 1 + 3.94T + 11T^{2} \)
13 \( 1 + (4.03 + 4.03i)T + 13iT^{2} \)
17 \( 1 + (-3.90 - 3.90i)T + 17iT^{2} \)
23 \( 1 + (0.537 - 0.537i)T - 23iT^{2} \)
29 \( 1 + 6.28T + 29T^{2} \)
31 \( 1 - 7.00iT - 31T^{2} \)
37 \( 1 + (1.05 - 1.05i)T - 37iT^{2} \)
41 \( 1 - 3.03iT - 41T^{2} \)
43 \( 1 + (5.70 - 5.70i)T - 43iT^{2} \)
47 \( 1 + (-2.04 - 2.04i)T + 47iT^{2} \)
53 \( 1 + (4.39 + 4.39i)T + 53iT^{2} \)
59 \( 1 - 2.32T + 59T^{2} \)
61 \( 1 + 9.32T + 61T^{2} \)
67 \( 1 + (7.35 - 7.35i)T - 67iT^{2} \)
71 \( 1 - 9.62iT - 71T^{2} \)
73 \( 1 + (-4.47 + 4.47i)T - 73iT^{2} \)
79 \( 1 - 0.991T + 79T^{2} \)
83 \( 1 + (6.64 - 6.64i)T - 83iT^{2} \)
89 \( 1 + 7.09T + 89T^{2} \)
97 \( 1 + (-7.11 + 7.11i)T - 97iT^{2} \)
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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

−9.884754379460353740008251921826, −8.534113081336856209628902689126, −7.88238412765694692623042782571, −7.44333931363298778944054533713, −5.92298922561051633503449507647, −5.37171608185935306752391198831, −4.73033008409051325862539237067, −3.32815585244585098946967519621, −2.75611213560698697752316617178, −1.68423575356467772194834956685, 0.44116787071862627035374925156, 1.97224761093616249287995330547, 3.42706719257000704437749303232, 4.48113354641885586501920956737, 4.85936045198969782536967563960, 5.54131287746248700209697751651, 7.21981609380822619319631579395, 7.46358953585498973810837100635, 7.932156834400943744176861110717, 9.016387780098316004983594912434

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