↑

Properties

Label 2-105-7.4-c1-0-3
Degree $2$
Conductor $105$
Sign $0.827 - 0.561i$
Analytic cond. $0.838429$
Root an. cond. $0.915657$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.707 + 1.22i)2-s + (0.5 − 0.866i)3-s + (0.5 + 0.866i)5-s + 1.41·6-s + (−2.62 + 0.358i)7-s + 2.82·8-s + (−0.499 − 0.866i)9-s + (−0.707 + 1.22i)10-s + (−0.292 + 0.507i)11-s − 4.41·13-s + (−2.29 − 2.95i)14-s + 0.999·15-s + (2.00 + 3.46i)16-s + (−1.12 + 1.94i)17-s + (0.707 − 1.22i)18-s + (−2.32 − 4.03i)19-s + ⋯
L(s)  = 1  + (0.499 + 0.866i)2-s + (0.288 − 0.499i)3-s + (0.223 + 0.387i)5-s + 0.577·6-s + (−0.990 + 0.135i)7-s + 0.999·8-s + (−0.166 − 0.288i)9-s + (−0.223 + 0.387i)10-s + (−0.0883 + 0.152i)11-s − 1.22·13-s + (−0.612 − 0.790i)14-s + 0.258·15-s + (0.500 + 0.866i)16-s + (−0.271 + 0.471i)17-s + (0.166 − 0.288i)18-s + (−0.534 − 0.925i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.827 - 0.561i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.827 - 0.561i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(105\)    =    \(3 \cdot 5 \cdot 7\)
Sign: $0.827 - 0.561i$
Analytic conductor: \(0.838429\)
Root analytic conductor: \(0.915657\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{105} (46, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 105,\ (\ :1/2),\ 0.827 - 0.561i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.32155 + 0.406075i\)
\(L(\frac12)\) \(\approx\) \(1.32155 + 0.406075i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.5 + 0.866i)T \)
5 \( 1 + (-0.5 - 0.866i)T \)
7 \( 1 + (2.62 - 0.358i)T \)
good2 \( 1 + (-0.707 - 1.22i)T + (-1 + 1.73i)T^{2} \)
11 \( 1 + (0.292 - 0.507i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 4.41T + 13T^{2} \)
17 \( 1 + (1.12 - 1.94i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.32 + 4.03i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.12 + 1.94i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 8.24T + 29T^{2} \)
31 \( 1 + (-2.91 + 5.04i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-4.20 - 7.28i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 6.24T + 41T^{2} \)
43 \( 1 + 7.58T + 43T^{2} \)
47 \( 1 + (-6.65 - 11.5i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (3.41 - 5.91i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-0.707 + 1.22i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.24 - 3.88i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-6.86 + 11.8i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 0.585T + 71T^{2} \)
73 \( 1 + (-6.03 + 10.4i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (3.32 + 5.76i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 2.58T + 83T^{2} \)
89 \( 1 + (-6.12 - 10.6i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 5.17T + 97T^{2} \)
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

−13.90101365170978259713531828194, −13.16570407819526337686978989569, −12.12910369073666454920983101763, −10.55694361278503675742509264830, −9.582192795867821887993136905510, −8.023002158108103590266103382858, −6.80204417439784926319119941577, −6.26845592511871665743518898927, −4.67396113575950758188924115996, −2.60443631793151104599636944450, 2.52482421687637334071804695949, 3.82006576500631858148861929335, 5.10864985383856050819856580269, 6.91256119582870925137041749674, 8.370373083576000785921806442735, 9.810283539703820155971404554952, 10.35536265762828884247062641334, 11.81671890009154902078871401893, 12.56339452330485471504790081671, 13.48740720179644072341838222314

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