Properties

Label 2-1170-39.2-c1-0-0
Degree $2$
Conductor $1170$
Sign $-0.124 - 0.992i$
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  + (0.258 − 0.965i)2-s + (−0.866 − 0.499i)4-s + (−0.707 − 0.707i)5-s + (−2.36 + 0.633i)7-s + (−0.707 + 0.707i)8-s + (−0.866 + 0.500i)10-s + (0.965 + 0.258i)11-s + (1 − 3.46i)13-s + 2.44i·14-s + (0.500 + 0.866i)16-s + (−2.63 + 4.57i)17-s + (−1 − 3.73i)19-s + (0.258 + 0.965i)20-s + (0.499 − 0.866i)22-s + (−0.258 − 0.448i)23-s + ⋯
L(s)  = 1  + (0.183 − 0.683i)2-s + (−0.433 − 0.249i)4-s + (−0.316 − 0.316i)5-s + (−0.894 + 0.239i)7-s + (−0.249 + 0.249i)8-s + (−0.273 + 0.158i)10-s + (0.291 + 0.0780i)11-s + (0.277 − 0.960i)13-s + 0.654i·14-s + (0.125 + 0.216i)16-s + (−0.640 + 1.10i)17-s + (−0.229 − 0.856i)19-s + (0.0578 + 0.215i)20-s + (0.106 − 0.184i)22-s + (−0.0539 − 0.0934i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.124 - 0.992i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.124 - 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1170\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 13\)
Sign: $-0.124 - 0.992i$
Analytic conductor: \(9.34249\)
Root analytic conductor: \(3.05654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1170} (431, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1170,\ (\ :1/2),\ -0.124 - 0.992i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2605653880\)
\(L(\frac12)\) \(\approx\) \(0.2605653880\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.258 + 0.965i)T \)
3 \( 1 \)
5 \( 1 + (0.707 + 0.707i)T \)
13 \( 1 + (-1 + 3.46i)T \)
good7 \( 1 + (2.36 - 0.633i)T + (6.06 - 3.5i)T^{2} \)
11 \( 1 + (-0.965 - 0.258i)T + (9.52 + 5.5i)T^{2} \)
17 \( 1 + (2.63 - 4.57i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1 + 3.73i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (0.258 + 0.448i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (8.03 - 4.64i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (7.09 - 7.09i)T - 31iT^{2} \)
37 \( 1 + (0.133 - 0.5i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + (2.82 - 10.5i)T + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (-10.7 - 6.23i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (3.53 - 3.53i)T - 47iT^{2} \)
53 \( 1 + 10.1iT - 53T^{2} \)
59 \( 1 + (-3.55 - 13.2i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (-5.83 + 10.0i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (4.09 + 1.09i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + (14.0 - 3.77i)T + (61.4 - 35.5i)T^{2} \)
73 \( 1 + (7.92 + 7.92i)T + 73iT^{2} \)
79 \( 1 - 5.92T + 79T^{2} \)
83 \( 1 + (7.72 + 7.72i)T + 83iT^{2} \)
89 \( 1 + (-9.84 - 2.63i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (-0.464 - 1.73i)T + (-84.0 + 48.5i)T^{2} \)
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

−10.07669049295936496961722856746, −9.142849176088523166695120018861, −8.688214735110433723263619847823, −7.60229640967944515186980939389, −6.53275621062522438190284549668, −5.71867311665207907807811001057, −4.70885554710915065793554455239, −3.68477756092872092522051321598, −2.94571933261758113809294604725, −1.53172959916422802530505201196, 0.10421374424249350858873641144, 2.23755416049680252510454332448, 3.71828638454772267439914895987, 4.10662378040484246762383772583, 5.52860366994167483619551562710, 6.25705709379293982091647822991, 7.14309209412780788474376988148, 7.56191813145756465922735097189, 8.898842213701976641256077358762, 9.304853819716815344416128595446

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