Properties

Label 2-1755-195.8-c0-0-1
Degree $2$
Conductor $1755$
Sign $-0.661 - 0.749i$
Analytic cond. $0.875859$
Root an. cond. $0.935873$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.41i·2-s − 1.00·4-s + (−0.707 − 0.707i)5-s i·7-s + (−1.00 + 1.00i)10-s + (−0.707 − 0.707i)11-s + i·13-s − 1.41·14-s − 0.999·16-s + (−0.707 + 0.707i)17-s + (1 − i)19-s + (0.707 + 0.707i)20-s + (−1.00 + 1.00i)22-s + 1.00i·25-s + 1.41·26-s + ⋯
L(s)  = 1  − 1.41i·2-s − 1.00·4-s + (−0.707 − 0.707i)5-s i·7-s + (−1.00 + 1.00i)10-s + (−0.707 − 0.707i)11-s + i·13-s − 1.41·14-s − 0.999·16-s + (−0.707 + 0.707i)17-s + (1 − i)19-s + (0.707 + 0.707i)20-s + (−1.00 + 1.00i)22-s + 1.00i·25-s + 1.41·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1755 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.661 - 0.749i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1755 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.661 - 0.749i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1755\)    =    \(3^{3} \cdot 5 \cdot 13\)
Sign: $-0.661 - 0.749i$
Analytic conductor: \(0.875859\)
Root analytic conductor: \(0.935873\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1755} (593, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1755,\ (\ :0),\ -0.661 - 0.749i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7113428581\)
\(L(\frac12)\) \(\approx\) \(0.7113428581\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 + (0.707 + 0.707i)T \)
13 \( 1 - iT \)
good2 \( 1 + 1.41iT - T^{2} \)
7 \( 1 + iT - T^{2} \)
11 \( 1 + (0.707 + 0.707i)T + iT^{2} \)
17 \( 1 + (0.707 - 0.707i)T - iT^{2} \)
19 \( 1 + (-1 + i)T - iT^{2} \)
23 \( 1 + iT^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 + (1 + i)T + iT^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + (-0.707 + 0.707i)T - iT^{2} \)
43 \( 1 + (1 - i)T - iT^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 + (-0.707 + 0.707i)T - iT^{2} \)
59 \( 1 + (-0.707 + 0.707i)T - iT^{2} \)
61 \( 1 - T + T^{2} \)
67 \( 1 + T + T^{2} \)
71 \( 1 + (0.707 - 0.707i)T - iT^{2} \)
73 \( 1 + T + T^{2} \)
79 \( 1 + iT - T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 + (1.41 - 1.41i)T - iT^{2} \)
97 \( 1 + T + T^{2} \)
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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

−9.155553299753020804272522128712, −8.479817485852909443401654106490, −7.47405667097760737642710297983, −6.80077281425422708493434812369, −5.42403926061772187696951895999, −4.36638692038019493248726688942, −3.94167618332493951033016245345, −2.96245342177771874046217349872, −1.72278458366729844328234675086, −0.52846324518906940756117331930, 2.33685078598129499902922578531, 3.24555355172147206812645862899, 4.57909120856820058183067336954, 5.43864154959947282944467900413, 5.92874984167280087492863516643, 7.18217898327931362595832054857, 7.31278896940280515320478139739, 8.249391397593048941380757687603, 8.804197382695310357694372605655, 9.854742386865629009854504018165

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