↑

Properties

Label 2-2340-39.11-c1-0-14
Degree $2$
Conductor $2340$
Sign $0.450 + 0.892i$
Analytic cond. $18.6849$
Root an. cond. $4.32261$
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)5-s + (0.0943 + 0.352i)7-s + (−0.741 + 2.76i)11-s + (2.55 − 2.54i)13-s + (2.67 − 4.63i)17-s + (−0.440 + 0.118i)19-s + (−4.23 − 7.33i)23-s − 1.00i·25-s + (−3.94 + 2.27i)29-s + (−5.97 − 5.97i)31-s + (0.315 + 0.182i)35-s + (8.63 + 2.31i)37-s + (4.92 + 1.31i)41-s + (3.85 + 2.22i)43-s + (1.40 + 1.40i)47-s + ⋯
L(s)  = 1  + (0.316 − 0.316i)5-s + (0.0356 + 0.133i)7-s + (−0.223 + 0.834i)11-s + (0.709 − 0.704i)13-s + (0.649 − 1.12i)17-s + (−0.101 + 0.0270i)19-s + (−0.883 − 1.52i)23-s − 0.200i·25-s + (−0.733 + 0.423i)29-s + (−1.07 − 1.07i)31-s + (0.0533 + 0.0308i)35-s + (1.41 + 0.380i)37-s + (0.769 + 0.206i)41-s + (0.587 + 0.339i)43-s + (0.204 + 0.204i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2340 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.450 + 0.892i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2340 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.450 + 0.892i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2340\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 13\)
Sign: $0.450 + 0.892i$
Analytic conductor: \(18.6849\)
Root analytic conductor: \(4.32261\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2340} (1961, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2340,\ (\ :1/2),\ 0.450 + 0.892i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.750521299\)
\(L(\frac12)\) \(\approx\) \(1.750521299\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 + (-0.707 + 0.707i)T \)
13 \( 1 + (-2.55 + 2.54i)T \)
good7 \( 1 + (-0.0943 - 0.352i)T + (-6.06 + 3.5i)T^{2} \)
11 \( 1 + (0.741 - 2.76i)T + (-9.52 - 5.5i)T^{2} \)
17 \( 1 + (-2.67 + 4.63i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.440 - 0.118i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + (4.23 + 7.33i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (3.94 - 2.27i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (5.97 + 5.97i)T + 31iT^{2} \)
37 \( 1 + (-8.63 - 2.31i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (-4.92 - 1.31i)T + (35.5 + 20.5i)T^{2} \)
43 \( 1 + (-3.85 - 2.22i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-1.40 - 1.40i)T + 47iT^{2} \)
53 \( 1 - 9.32iT - 53T^{2} \)
59 \( 1 + (5.14 - 1.37i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (2.66 - 4.61i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.61 + 13.4i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + (3.70 + 13.8i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 + (-6.70 + 6.70i)T - 73iT^{2} \)
79 \( 1 - 8.79T + 79T^{2} \)
83 \( 1 + (-5.75 + 5.75i)T - 83iT^{2} \)
89 \( 1 + (-3.42 + 12.7i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (-6.07 + 1.62i)T + (84.0 - 48.5i)T^{2} \)
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

−9.100677452747049099646371427110, −7.80744668392588431705177858836, −7.65435023264140713733208964401, −6.34996286482349012959022998216, −5.77752118557059126061894774680, −4.87019291799388365694346027848, −4.10641521373208610603092840706, −2.91613338452432162459395807885, −2.01399444909631802321436216591, −0.64416070819171608231152311217, 1.24532281069175602908983641557, 2.28109230840494915312204234422, 3.60100796607920125700732843022, 3.98126427509455692197513090822, 5.54368636197156052055714260885, 5.82419542745006597304292284145, 6.76267926721003460893402218763, 7.66825320136925526376957189914, 8.297892642554572758115143103345, 9.193442739643819667842015775813

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