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Properties

Label 2-1710-95.37-c1-0-9
Degree $2$
Conductor $1710$
Sign $-0.960 - 0.277i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 1710 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.960 - 0.277i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1710 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.960 - 0.277i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1710\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 19\)
Sign: $-0.960 - 0.277i$
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: \((2,\ 1710,\ (\ :1/2),\ -0.960 - 0.277i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.139515073\)
\(L(\frac12)\) \(\approx\) \(1.139515073\)
\(L(\frac{3}{2})\) 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

−9.555498652994606628187859522151, −8.544993324646561733631343395387, −8.125593444668772179177232889367, −7.66618058496381582226911952912, −6.34960095324868931973357248567, −5.92084063898364521880116644082, −4.88434721380291279575275254557, −3.92733384977914062130603503619, −2.57131195463676447418860621009, −1.63680926923515856458952302684, 0.56896372286145013399843839109, 1.33113088210507698080741066215, 2.90189030879461307684134856830, 3.87913253792855836639476326536, 4.80817427621996908165159666206, 5.34340659838982725104891767791, 6.96040847911230761851706851261, 7.69844849033927213765810579830, 8.306601652205821177074210100002, 8.603198172213119561697294767184

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