Properties

Label 2-1755-9.4-c1-0-23
Degree $2$
Conductor $1755$
Sign $-0.847 + 0.531i$
Analytic cond. $14.0137$
Root an. cond. $3.74349$
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  + (−0.984 − 1.70i)2-s + (−0.938 + 1.62i)4-s + (−0.5 + 0.866i)5-s + (1.51 + 2.62i)7-s − 0.240·8-s + 1.96·10-s + (−2.15 − 3.72i)11-s + (0.5 − 0.866i)13-s + (2.98 − 5.16i)14-s + (2.11 + 3.66i)16-s + 0.303·17-s − 6.04·19-s + (−0.938 − 1.62i)20-s + (−4.23 + 7.33i)22-s + (1.47 − 2.55i)23-s + ⋯
L(s)  = 1  + (−0.696 − 1.20i)2-s + (−0.469 + 0.813i)4-s + (−0.223 + 0.387i)5-s + (0.572 + 0.991i)7-s − 0.0850·8-s + 0.622·10-s + (−0.648 − 1.12i)11-s + (0.138 − 0.240i)13-s + (0.796 − 1.38i)14-s + (0.528 + 0.915i)16-s + 0.0735·17-s − 1.38·19-s + (−0.209 − 0.363i)20-s + (−0.903 + 1.56i)22-s + (0.307 − 0.532i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1755 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.847 + 0.531i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1755 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.847 + 0.531i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1755\)    =    \(3^{3} \cdot 5 \cdot 13\)
Sign: $-0.847 + 0.531i$
Analytic conductor: \(14.0137\)
Root analytic conductor: \(3.74349\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1755} (1171, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1755,\ (\ :1/2),\ -0.847 + 0.531i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7824530845\)
\(L(\frac12)\) \(\approx\) \(0.7824530845\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 + (0.5 - 0.866i)T \)
13 \( 1 + (-0.5 + 0.866i)T \)
good2 \( 1 + (0.984 + 1.70i)T + (-1 + 1.73i)T^{2} \)
7 \( 1 + (-1.51 - 2.62i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (2.15 + 3.72i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 - 0.303T + 17T^{2} \)
19 \( 1 + 6.04T + 19T^{2} \)
23 \( 1 + (-1.47 + 2.55i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-4.44 - 7.69i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (0.0319 - 0.0554i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 11.1T + 37T^{2} \)
41 \( 1 + (-4.09 + 7.09i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (1.45 + 2.52i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (3.44 + 5.96i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 11.8T + 53T^{2} \)
59 \( 1 + (-4.57 + 7.92i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (0.657 + 1.13i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-7.91 + 13.7i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 2.85T + 71T^{2} \)
73 \( 1 + 11.2T + 73T^{2} \)
79 \( 1 + (5.17 + 8.97i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (3.53 + 6.12i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 4.26T + 89T^{2} \)
97 \( 1 + (2.91 + 5.05i)T + (-48.5 + 84.0i)T^{2} \)
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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

−8.794517480458053883309018951548, −8.605884682184950997791171983349, −7.83783282381642417385698496439, −6.48051031768131699135325589261, −5.78105291509093637015956329770, −4.74338090630136804534391124499, −3.44325248866401318257599282054, −2.72924005819586275139295504050, −1.90082155146537686678607695298, −0.42921168993770810393607736437, 1.08579486762054996185415522657, 2.60054806991606079674799815422, 4.28019727394816761743451883736, 4.64788798488531291867633559738, 5.87083847295876220647289941691, 6.63020953551486668290869204012, 7.47824051403303779801232428873, 7.919177502126785634803966752948, 8.501748816861754953918589228230, 9.589537437745057402599251411421

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