↑

Properties

Label 2-1170-195.164-c1-0-14
Degree $2$
Conductor $1170$
Sign $0.186 - 0.982i$
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 + (2.23 − 0.137i)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 + (0.137 + 2.23i)20-s + 3.74i·22-s + 6.48i·23-s + ⋯
L(s)  = 1  + (0.499 + 0.499i)2-s + 0.500i·4-s + (0.998 − 0.0613i)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 + (0.0306 + 0.499i)20-s + 0.797i·22-s + 1.35i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.186 - 0.982i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.186 - 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1170\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 13\)
Sign: $0.186 - 0.982i$
Analytic conductor: \(9.34249\)
Root analytic conductor: \(3.05654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1170} (359, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1170,\ (\ :1/2),\ 0.186 - 0.982i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.785057272\)
\(L(\frac12)\) \(\approx\) \(2.785057272\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.23 + 0.137i)T \)
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

−9.610872421976227775303349292839, −9.141594998861294271419588591350, −8.467286896312769225269673065503, −7.09430660143787325730890098958, −6.72848459549212877129459557982, −5.62639263174869691134361428121, −4.95120594621007727991185719462, −4.14075209447003455053241222830, −2.56804059465085655853103634058, −1.74973701358156747661828042313, 1.14738413981353760019845251894, 2.08148101028668024093911907512, 3.47174957769962860787942769740, 4.23572957015638280435398604879, 5.36601979197871661729778178803, 6.12287963652810062640645233689, 6.78359527464775112202999716697, 8.346959445826065481242215525955, 8.561992190760566902187694332983, 9.986592363490305299983018171623

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