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Properties

Label 2-1755-65.38-c0-0-3
Degree $2$
Conductor $1755$
Sign $0.354 + 0.934i$
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  + (−1.12 − 1.12i)2-s + 1.51i·4-s + (0.793 + 0.608i)5-s + (0.580 − 0.580i)8-s + (−0.207 − 1.57i)10-s − 1.84i·11-s + (0.707 + 0.707i)13-s + 0.214·16-s + (−0.923 + 1.20i)20-s + (−2.07 + 2.07i)22-s + (0.258 + 0.965i)25-s − 1.58i·26-s + (−0.821 − 0.821i)32-s + (0.814 − 0.107i)40-s + 0.765i·41-s + ⋯
L(s)  = 1  + (−1.12 − 1.12i)2-s + 1.51i·4-s + (0.793 + 0.608i)5-s + (0.580 − 0.580i)8-s + (−0.207 − 1.57i)10-s − 1.84i·11-s + (0.707 + 0.707i)13-s + 0.214·16-s + (−0.923 + 1.20i)20-s + (−2.07 + 2.07i)22-s + (0.258 + 0.965i)25-s − 1.58i·26-s + (−0.821 − 0.821i)32-s + (0.814 − 0.107i)40-s + 0.765i·41-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1755\)    =    \(3^{3} \cdot 5 \cdot 13\)
Sign: $0.354 + 0.934i$
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: \((2,\ 1755,\ (\ :0),\ 0.354 + 0.934i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7466900465\)
\(L(\frac12)\) \(\approx\) \(0.7466900465\)
\(L(1)\) 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 \( 1 + (-0.793 - 0.608i)T \)
13 \( 1 + (-0.707 - 0.707i)T \)
good2 \( 1 + (1.12 + 1.12i)T + iT^{2} \)
7 \( 1 + iT^{2} \)
11 \( 1 + 1.84iT - T^{2} \)
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 \( 1 - 0.765iT - T^{2} \)
43 \( 1 + (0.366 + 0.366i)T + iT^{2} \)
47 \( 1 + (-0.860 - 0.860i)T + iT^{2} \)
53 \( 1 + iT^{2} \)
59 \( 1 - 1.98T + T^{2} \)
61 \( 1 - 1.93T + T^{2} \)
67 \( 1 + iT^{2} \)
71 \( 1 + 1.21iT - T^{2} \)
73 \( 1 - iT^{2} \)
79 \( 1 + 1.41iT - T^{2} \)
83 \( 1 + (1.40 - 1.40i)T - iT^{2} \)
89 \( 1 - 0.261T + T^{2} \)
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

−9.460406625447310480546484768049, −8.666247357788741388018594496265, −8.295116120921050728536877574367, −7.06795946658242681038396123906, −6.17162501061736257454812148411, −5.48026839261023035524735304497, −3.80805955201980304307891958856, −3.09281131542526592104136575117, −2.17396076894382506558321814956, −1.05994885382981736182188441304, 1.16440633105369252653000682430, 2.31255464490977094334701274190, 4.03347597476201547814085539842, 5.17653117528945443069448631186, 5.73128183772959659681666089809, 6.72650533776244787915397087089, 7.24627757753798095692376687206, 8.201007038694026574656369005930, 8.729768695244601398368667012312, 9.587879429751302856887901000177

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