↑

Properties

Label 2-1170-195.44-c1-0-4
Degree $2$
Conductor $1170$
Sign $-0.952 + 0.305i$
Analytic cond. $9.34249$
Root an. cond. $3.05654$
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 + 0.707i)2-s − 1.00i·4-s + (0.137 + 2.23i)5-s + (−1.96 + 1.96i)7-s + (0.707 + 0.707i)8-s + (−1.67 − 1.48i)10-s + (2.64 − 2.64i)11-s + (−0.770 + 3.52i)13-s − 2.77i·14-s − 1.00·16-s − 5.46i·17-s + (−5.64 + 5.64i)19-s + (2.23 − 0.137i)20-s + 3.74i·22-s + 6.48i·23-s + ⋯
L(s)  = 1  + (−0.499 + 0.499i)2-s − 0.500i·4-s + (0.0613 + 0.998i)5-s + (−0.742 + 0.742i)7-s + (0.250 + 0.250i)8-s + (−0.529 − 0.468i)10-s + (0.797 − 0.797i)11-s + (−0.213 + 0.976i)13-s − 0.742i·14-s − 0.250·16-s − 1.32i·17-s + (−1.29 + 1.29i)19-s + (0.499 − 0.0306i)20-s + 0.797i·22-s + 1.35i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.952 + 0.305i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.952 + 0.305i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1170\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 13\)
Sign: $-0.952 + 0.305i$
Analytic conductor: \(9.34249\)
Root analytic conductor: \(3.05654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1170} (629, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1170,\ (\ :1/2),\ -0.952 + 0.305i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5414051445\)
\(L(\frac12)\) \(\approx\) \(0.5414051445\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.707 - 0.707i)T \)
3 \( 1 \)
5 \( 1 + (-0.137 - 2.23i)T \)
13 \( 1 + (0.770 - 3.52i)T \)
good7 \( 1 + (1.96 - 1.96i)T - 7iT^{2} \)
11 \( 1 + (-2.64 + 2.64i)T - 11iT^{2} \)
17 \( 1 + 5.46iT - 17T^{2} \)
19 \( 1 + (5.64 - 5.64i)T - 19iT^{2} \)
23 \( 1 - 6.48iT - 23T^{2} \)
29 \( 1 - 2.74iT - 29T^{2} \)
31 \( 1 + (-0.945 + 0.945i)T - 31iT^{2} \)
37 \( 1 + (-6.10 + 6.10i)T - 37iT^{2} \)
41 \( 1 + (5.19 + 5.19i)T + 41iT^{2} \)
43 \( 1 + 1.08T + 43T^{2} \)
47 \( 1 + (-0.108 - 0.108i)T + 47iT^{2} \)
53 \( 1 + 12.5T + 53T^{2} \)
59 \( 1 + (-9.67 + 9.67i)T - 59iT^{2} \)
61 \( 1 + 11.6T + 61T^{2} \)
67 \( 1 + (4.45 + 4.45i)T + 67iT^{2} \)
71 \( 1 + (-8.97 - 8.97i)T + 71iT^{2} \)
73 \( 1 + (7.14 - 7.14i)T - 73iT^{2} \)
79 \( 1 + 17.7T + 79T^{2} \)
83 \( 1 + (1.82 - 1.82i)T - 83iT^{2} \)
89 \( 1 + (-2.49 + 2.49i)T - 89iT^{2} \)
97 \( 1 + (0.837 + 0.837i)T + 97iT^{2} \)
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

−9.912185160900545862409320286893, −9.422753746574716791248322431200, −8.722815867160391523927568095882, −7.66998901785320292713096004122, −6.81275705815568092662388355383, −6.24654915503892389021254174266, −5.53917076531415168976753803528, −4.03921778409641983136732692817, −3.04408121099930846851494835368, −1.85765848493159514014033742062, 0.27327544779496697662048845862, 1.53764432997862317078783800248, 2.86739271832808791823658528554, 4.18470482864569868397521338112, 4.61480243690928878412611741942, 6.17451270273036544636408904722, 6.81708973858486900211072986165, 7.984020035836678346136958011473, 8.576477893553896408484314566407, 9.407339880349982112513499988644

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