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Properties

Label 2-1755-9.4-c1-0-1
Degree $2$
Conductor $1755$
Sign $-0.180 + 0.983i$
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  + (1.12 + 1.95i)2-s + (−1.54 + 2.67i)4-s + (−0.5 + 0.866i)5-s + (−0.353 − 0.611i)7-s − 2.46·8-s − 2.25·10-s + (−1.16 − 2.01i)11-s + (0.5 − 0.866i)13-s + (0.797 − 1.38i)14-s + (0.311 + 0.538i)16-s − 6.04·17-s − 7.91·19-s + (−1.54 − 2.67i)20-s + (2.62 − 4.55i)22-s + (−4.47 + 7.75i)23-s + ⋯
L(s)  = 1  + (0.797 + 1.38i)2-s + (−0.773 + 1.33i)4-s + (−0.223 + 0.387i)5-s + (−0.133 − 0.231i)7-s − 0.871·8-s − 0.713·10-s + (−0.351 − 0.608i)11-s + (0.138 − 0.240i)13-s + (0.213 − 0.369i)14-s + (0.0777 + 0.134i)16-s − 1.46·17-s − 1.81·19-s + (−0.345 − 0.598i)20-s + (0.560 − 0.970i)22-s + (−0.933 + 1.61i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1755 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.180 + 0.983i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1755 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.180 + 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1755\)    =    \(3^{3} \cdot 5 \cdot 13\)
Sign: $-0.180 + 0.983i$
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: \((2,\ 1755,\ (\ :1/2),\ -0.180 + 0.983i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6446166731\)
\(L(\frac12)\) \(\approx\) \(0.6446166731\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.12 - 1.95i)T + (-1 + 1.73i)T^{2} \)
7 \( 1 + (0.353 + 0.611i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (1.16 + 2.01i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + 6.04T + 17T^{2} \)
19 \( 1 + 7.91T + 19T^{2} \)
23 \( 1 + (4.47 - 7.75i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.20 + 2.08i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-1.53 + 2.65i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 2.38T + 37T^{2} \)
41 \( 1 + (4.72 - 8.19i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (5.96 + 10.3i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-4.86 - 8.42i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 5.90T + 53T^{2} \)
59 \( 1 + (-1.48 + 2.57i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4.62 - 8.00i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.847 + 1.46i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 12.4T + 71T^{2} \)
73 \( 1 - 4.88T + 73T^{2} \)
79 \( 1 + (-5.51 - 9.54i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (0.260 + 0.451i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 10.7T + 89T^{2} \)
97 \( 1 + (-1.48 - 2.56i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.773059187703314748833570971252, −8.572947351290267911542785945788, −8.182396159257339137316170078937, −7.29075381640367438723959066462, −6.55133529326915185460176854812, −6.03764202454914874750826291838, −5.13476333232777436755637445649, −4.16702456103955426100935327970, −3.59004681119306719859611618918, −2.17238418427712977316186118558, 0.16625902607556915075534384096, 1.93026413047034764827403929097, 2.40202466608635142687440146796, 3.70943328457767098422031929091, 4.47042221316678574959087976455, 4.92509759702577389612765934146, 6.18359729718260267940035700015, 6.92831759929359998715435087725, 8.291686802024559312459900166083, 8.808185836909643424457773263864

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