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Properties

Label 2-1755-195.194-c0-0-4
Degree $2$
Conductor $1755$
Sign $0.258 - 0.965i$
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  + 0.517i·2-s + 0.732·4-s + (−0.258 + 0.965i)5-s + 0.896i·8-s + (−0.499 − 0.133i)10-s + 1.41·11-s − i·13-s + 0.267·16-s + (−0.189 + 0.707i)20-s + 0.732i·22-s + (−0.866 − 0.499i)25-s + 0.517·26-s + 1.03i·32-s + (−0.866 − 0.232i)40-s − 1.41·41-s + ⋯
L(s)  = 1  + 0.517i·2-s + 0.732·4-s + (−0.258 + 0.965i)5-s + 0.896i·8-s + (−0.499 − 0.133i)10-s + 1.41·11-s − i·13-s + 0.267·16-s + (−0.189 + 0.707i)20-s + 0.732i·22-s + (−0.866 − 0.499i)25-s + 0.517·26-s + 1.03i·32-s + (−0.866 − 0.232i)40-s − 1.41·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1755 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.258 - 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1755 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.258 - 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1755\)    =    \(3^{3} \cdot 5 \cdot 13\)
Sign: $0.258 - 0.965i$
Analytic conductor: \(0.875859\)
Root analytic conductor: \(0.935873\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1755} (1754, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1755,\ (\ :0),\ 0.258 - 0.965i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.413872651\)
\(L(\frac12)\) \(\approx\) \(1.413872651\)
\(L(1)\) 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 \( 1 + (0.258 - 0.965i)T \)
13 \( 1 + iT \)
good2 \( 1 - 0.517iT - T^{2} \)
7 \( 1 + T^{2} \)
11 \( 1 - 1.41T + T^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 + 1.41T + T^{2} \)
43 \( 1 - 1.73iT - T^{2} \)
47 \( 1 + 0.517iT - T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 + 1.93T + T^{2} \)
61 \( 1 - 1.73T + T^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 + 0.517T + T^{2} \)
73 \( 1 + T^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 + 1.93iT - T^{2} \)
89 \( 1 - 1.93T + T^{2} \)
97 \( 1 + T^{2} \)
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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

−9.761461361560952284256633447542, −8.656963087909443034781708771096, −7.88347173625372159447299603365, −7.21685940415650701967351193554, −6.44772935546086059491407255225, −6.03138793770543933808735587014, −4.85988355361324855615640988501, −3.59224778664156823075575678562, −2.91289431483528540178095404160, −1.66451695965392663373453294438, 1.24992497373150315766066356298, 2.04806880589301487567083656228, 3.51712617256249690639391528431, 4.12189994980774336335384652054, 5.14065860330983750885210510563, 6.30575300467800886445988850322, 6.83437660141741352850221526785, 7.76319985146909705677246256588, 8.769621908684172268269110171720, 9.301856879864563916469737963572

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