↑

Properties

Label 2-105-105.62-c2-0-25
Degree $2$
Conductor $105$
Sign $-0.139 + 0.990i$
Analytic cond. $2.86104$
Root an. cond. $1.69146$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (1.67 − 1.67i)2-s + (1.56 − 2.55i)3-s − 1.58i·4-s + (−0.529 − 4.97i)5-s + (−1.66 − 6.89i)6-s + (−1.73 + 6.78i)7-s + (4.03 + 4.03i)8-s + (−4.10 − 8.01i)9-s + (−9.19 − 7.42i)10-s + 4.41i·11-s + (−4.06 − 2.48i)12-s + (−1.62 − 1.62i)13-s + (8.43 + 14.2i)14-s + (−13.5 − 6.42i)15-s + 19.8·16-s + (13.9 + 13.9i)17-s + ⋯
L(s)  = 1  + (0.835 − 0.835i)2-s + (0.521 − 0.853i)3-s − 0.397i·4-s + (−0.105 − 0.994i)5-s + (−0.277 − 1.14i)6-s + (−0.247 + 0.968i)7-s + (0.503 + 0.503i)8-s + (−0.455 − 0.890i)9-s + (−0.919 − 0.742i)10-s + 0.401i·11-s + (−0.338 − 0.207i)12-s + (−0.124 − 0.124i)13-s + (0.602 + 1.01i)14-s + (−0.903 − 0.428i)15-s + 1.23·16-s + (0.819 + 0.819i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.139 + 0.990i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.139 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(105\)    =    \(3 \cdot 5 \cdot 7\)
Sign: $-0.139 + 0.990i$
Analytic conductor: \(2.86104\)
Root analytic conductor: \(1.69146\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{105} (62, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 105,\ (\ :1),\ -0.139 + 0.990i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.47966 - 1.70245i\)
\(L(\frac12)\) \(\approx\) \(1.47966 - 1.70245i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.56 + 2.55i)T \)
5 \( 1 + (0.529 + 4.97i)T \)
7 \( 1 + (1.73 - 6.78i)T \)
good2 \( 1 + (-1.67 + 1.67i)T - 4iT^{2} \)
11 \( 1 - 4.41iT - 121T^{2} \)
13 \( 1 + (1.62 + 1.62i)T + 169iT^{2} \)
17 \( 1 + (-13.9 - 13.9i)T + 289iT^{2} \)
19 \( 1 - 0.694T + 361T^{2} \)
23 \( 1 + (23.1 + 23.1i)T + 529iT^{2} \)
29 \( 1 - 49.1T + 841T^{2} \)
31 \( 1 - 33.8iT - 961T^{2} \)
37 \( 1 + (-2.02 - 2.02i)T + 1.36e3iT^{2} \)
41 \( 1 + 32.5T + 1.68e3T^{2} \)
43 \( 1 + (30.4 - 30.4i)T - 1.84e3iT^{2} \)
47 \( 1 + (-18.7 - 18.7i)T + 2.20e3iT^{2} \)
53 \( 1 + (33.8 + 33.8i)T + 2.80e3iT^{2} \)
59 \( 1 - 23.1iT - 3.48e3T^{2} \)
61 \( 1 + 12.9iT - 3.72e3T^{2} \)
67 \( 1 + (56.3 + 56.3i)T + 4.48e3iT^{2} \)
71 \( 1 + 92.7iT - 5.04e3T^{2} \)
73 \( 1 + (95.4 + 95.4i)T + 5.32e3iT^{2} \)
79 \( 1 - 100. iT - 6.24e3T^{2} \)
83 \( 1 + (5.62 - 5.62i)T - 6.88e3iT^{2} \)
89 \( 1 + 158. iT - 7.92e3T^{2} \)
97 \( 1 + (-37.1 + 37.1i)T - 9.40e3iT^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.90087458942267088485552868521, −12.27998874031359550413512604073, −11.93265651261624920498487634717, −10.12816809599837963630352588795, −8.678791550075977325475584375155, −7.972423587246759498131881507869, −6.13241668621429604618516500469, −4.75059367091699737687376638399, −3.16666648905723071461841728698, −1.75444612475016349392683330174, 3.25629723503058327957595516494, 4.28620890854881712169962089585, 5.71466911733367848460438396569, 7.01883452754972802378730696200, 7.932213019707104613863001938551, 9.841846660080081823962024260386, 10.35724155259221288417256453467, 11.65494203485059990203505368397, 13.56232163317423524747875067197, 13.94254936673558220135547417567

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