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Properties

Label 2-1755-65.38-c0-0-6
Degree $2$
Conductor $1755$
Sign $0.934 - 0.354i$
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  + (0.860 + 0.860i)2-s + 0.482i·4-s + (0.608 − 0.793i)5-s + (0.445 − 0.445i)8-s + (1.20 − 0.158i)10-s + 0.765i·11-s + (−0.707 − 0.707i)13-s + 1.24·16-s + (0.382 + 0.293i)20-s + (−0.658 + 0.658i)22-s + (−0.258 − 0.965i)25-s − 1.21i·26-s + (0.630 + 0.630i)32-s + (−0.0822 − 0.624i)40-s + 1.84i·41-s + ⋯
L(s)  = 1  + (0.860 + 0.860i)2-s + 0.482i·4-s + (0.608 − 0.793i)5-s + (0.445 − 0.445i)8-s + (1.20 − 0.158i)10-s + 0.765i·11-s + (−0.707 − 0.707i)13-s + 1.24·16-s + (0.382 + 0.293i)20-s + (−0.658 + 0.658i)22-s + (−0.258 − 0.965i)25-s − 1.21i·26-s + (0.630 + 0.630i)32-s + (−0.0822 − 0.624i)40-s + 1.84i·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1755 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.934 - 0.354i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1755 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.934 - 0.354i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1755\)    =    \(3^{3} \cdot 5 \cdot 13\)
Sign: $0.934 - 0.354i$
Analytic conductor: \(0.875859\)
Root analytic conductor: \(0.935873\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1755} (298, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1755,\ (\ :0),\ 0.934 - 0.354i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.976379440\)
\(L(\frac12)\) \(\approx\) \(1.976379440\)
\(L(1)\) 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 \( 1 + (-0.608 + 0.793i)T \)
13 \( 1 + (0.707 + 0.707i)T \)
good2 \( 1 + (-0.860 - 0.860i)T + iT^{2} \)
7 \( 1 + iT^{2} \)
11 \( 1 - 0.765iT - T^{2} \)
17 \( 1 - iT^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 + iT^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + iT^{2} \)
41 \( 1 - 1.84iT - T^{2} \)
43 \( 1 + (0.366 + 0.366i)T + iT^{2} \)
47 \( 1 + (-1.12 - 1.12i)T + iT^{2} \)
53 \( 1 + iT^{2} \)
59 \( 1 - 0.261T + T^{2} \)
61 \( 1 + 1.93T + T^{2} \)
67 \( 1 + iT^{2} \)
71 \( 1 - 1.58iT - T^{2} \)
73 \( 1 - iT^{2} \)
79 \( 1 - 1.41iT - T^{2} \)
83 \( 1 + (-0.184 + 0.184i)T - iT^{2} \)
89 \( 1 + 1.98T + T^{2} \)
97 \( 1 + iT^{2} \)
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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

−9.771490705497110672374598562013, −8.585288217400419781561342222340, −7.76032857710566221301552037962, −7.03226277076284229903998023026, −6.15989438456326051230104769440, −5.43875255815047401819553432056, −4.80920340215690177300347216752, −4.12685526180233811924501141799, −2.72300084096474299505063825070, −1.37641259683929665212270201746, 1.76534833526587080631205566889, 2.61745362105258369284024065838, 3.40625719191114367522039562647, 4.30403495990947505499187530028, 5.28941837355827788431449181226, 6.01784759878842613409551871949, 6.99359141711010862673817353090, 7.74570554926791551713264967093, 8.852285263427887165682527193572, 9.616137694193932725454373848741

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