↑

Properties

Label 2-1232-7.4-c1-0-11
Degree $2$
Conductor $1232$
Sign $-0.653 - 0.756i$
Analytic cond. $9.83756$
Root an. cond. $3.13649$
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  + (−1.20 + 2.08i)3-s + (−0.0393 − 0.0680i)5-s + (0.659 + 2.56i)7-s + (−1.39 − 2.41i)9-s + (0.5 − 0.866i)11-s + 6.30·13-s + 0.189·15-s + (−0.622 + 1.07i)17-s + (3.04 + 5.27i)19-s + (−6.13 − 1.70i)21-s + (3.85 + 6.67i)23-s + (2.49 − 4.32i)25-s − 0.499·27-s − 7.42·29-s + (4.54 − 7.86i)31-s + ⋯
L(s)  = 1  + (−0.694 + 1.20i)3-s + (−0.0175 − 0.0304i)5-s + (0.249 + 0.968i)7-s + (−0.465 − 0.806i)9-s + (0.150 − 0.261i)11-s + 1.74·13-s + 0.0488·15-s + (−0.150 + 0.261i)17-s + (0.698 + 1.20i)19-s + (−1.33 − 0.372i)21-s + (0.804 + 1.39i)23-s + (0.499 − 0.864i)25-s − 0.0961·27-s − 1.37·29-s + (0.815 − 1.41i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.653 - 0.756i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1232 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.653 - 0.756i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1232\)    =    \(2^{4} \cdot 7 \cdot 11\)
Sign: $-0.653 - 0.756i$
Analytic conductor: \(9.83756\)
Root analytic conductor: \(3.13649\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1232} (529, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1232,\ (\ :1/2),\ -0.653 - 0.756i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.364378879\)
\(L(\frac12)\) \(\approx\) \(1.364378879\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 + (-0.659 - 2.56i)T \)
11 \( 1 + (-0.5 + 0.866i)T \)
good3 \( 1 + (1.20 - 2.08i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (0.0393 + 0.0680i)T + (-2.5 + 4.33i)T^{2} \)
13 \( 1 - 6.30T + 13T^{2} \)
17 \( 1 + (0.622 - 1.07i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.04 - 5.27i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-3.85 - 6.67i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 7.42T + 29T^{2} \)
31 \( 1 + (-4.54 + 7.86i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-5.35 - 9.27i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 1.00T + 41T^{2} \)
43 \( 1 + 4.51T + 43T^{2} \)
47 \( 1 + (2.02 + 3.51i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (1.70 - 2.96i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (0.288 - 0.499i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (1.64 + 2.85i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.11 - 5.40i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 14.7T + 71T^{2} \)
73 \( 1 + (-5.41 + 9.38i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (3.78 + 6.56i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 6.19T + 83T^{2} \)
89 \( 1 + (-5.12 - 8.88i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 4.42T + 97T^{2} \)
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

−9.978465281092251575540216396006, −9.335487379557342352567642613395, −8.545922492450667220798521145034, −7.76596421421097099828498396664, −6.14728671280260382660764446770, −5.87988282338584346622443192050, −4.97035231650118449774456823707, −3.98448863632086657862092476789, −3.19661301028412371385706025405, −1.47781488843022610408755368634, 0.73419865070802353160128881869, 1.52631737983290850761782356126, 3.10145858076301892618902567792, 4.29566281885245783544490007937, 5.31032388982629104281484517656, 6.32539322652762848590703667435, 6.96588323954473905079191710252, 7.45716289284923332420315102402, 8.520401873592218163350539603227, 9.283018965084971775776150825996

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