↑

Properties

Label 2-416-52.23-c2-0-5
Degree $2$
Conductor $416$
Sign $-0.860 + 0.509i$
Analytic cond. $11.3351$
Root an. cond. $3.36677$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−2.83 + 1.63i)3-s + 5.42i·5-s + (−6.16 + 10.6i)7-s + (0.845 − 1.46i)9-s + (8.21 + 14.2i)11-s + (3.06 − 12.6i)13-s + (−8.87 − 15.3i)15-s + (−0.116 + 0.202i)17-s + (1.36 − 2.36i)19-s − 40.3i·21-s + (−16.5 + 9.54i)23-s − 4.45·25-s − 23.8i·27-s + (15.7 + 27.1i)29-s − 41.5·31-s + ⋯
L(s)  = 1  + (−0.943 + 0.544i)3-s + 1.08i·5-s + (−0.880 + 1.52i)7-s + (0.0939 − 0.162i)9-s + (0.746 + 1.29i)11-s + (0.235 − 0.971i)13-s + (−0.591 − 1.02i)15-s + (−0.00686 + 0.0118i)17-s + (0.0718 − 0.124i)19-s − 1.91i·21-s + (−0.718 + 0.415i)23-s − 0.178·25-s − 0.885i·27-s + (0.541 + 0.937i)29-s − 1.34·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.860 + 0.509i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.860 + 0.509i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(416\)    =    \(2^{5} \cdot 13\)
Sign: $-0.860 + 0.509i$
Analytic conductor: \(11.3351\)
Root analytic conductor: \(3.36677\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{416} (127, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 416,\ (\ :1),\ -0.860 + 0.509i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.188516 - 0.687960i\)
\(L(\frac12)\) \(\approx\) \(0.188516 - 0.687960i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
13 \( 1 + (-3.06 + 12.6i)T \)
good3 \( 1 + (2.83 - 1.63i)T + (4.5 - 7.79i)T^{2} \)
5 \( 1 - 5.42iT - 25T^{2} \)
7 \( 1 + (6.16 - 10.6i)T + (-24.5 - 42.4i)T^{2} \)
11 \( 1 + (-8.21 - 14.2i)T + (-60.5 + 104. i)T^{2} \)
17 \( 1 + (0.116 - 0.202i)T + (-144.5 - 250. i)T^{2} \)
19 \( 1 + (-1.36 + 2.36i)T + (-180.5 - 312. i)T^{2} \)
23 \( 1 + (16.5 - 9.54i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + (-15.7 - 27.1i)T + (-420.5 + 728. i)T^{2} \)
31 \( 1 + 41.5T + 961T^{2} \)
37 \( 1 + (-46.3 + 26.7i)T + (684.5 - 1.18e3i)T^{2} \)
41 \( 1 + (13.3 - 7.70i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (-39.8 - 22.9i)T + (924.5 + 1.60e3i)T^{2} \)
47 \( 1 - 11.3T + 2.20e3T^{2} \)
53 \( 1 + 93.5T + 2.80e3T^{2} \)
59 \( 1 + (-33.7 + 58.3i)T + (-1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (9.11 - 15.7i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (44.2 + 76.6i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + (-8.84 + 15.3i)T + (-2.52e3 - 4.36e3i)T^{2} \)
73 \( 1 - 30.0iT - 5.32e3T^{2} \)
79 \( 1 - 114. iT - 6.24e3T^{2} \)
83 \( 1 - 68.8T + 6.88e3T^{2} \)
89 \( 1 + (-135. + 78.0i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + (-42.7 - 24.6i)T + (4.70e3 + 8.14e3i)T^{2} \)
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

−11.41958142226962835930834827994, −10.67353470837427776734595134549, −9.810759721686086278531958859210, −9.147841696845182283529123121820, −7.69520940631416076672316924503, −6.51565161630256396376456941148, −5.92818126887638291841427673043, −4.95524773096306310025490475060, −3.47348050488229897990635429307, −2.33382248026562475914367962135, 0.39341446533279907309756620053, 1.19468627519203171591762084106, 3.61790535453625066093914382396, 4.49617950757904822708102973995, 5.97770386269216338887686754346, 6.45632647144829397318066659338, 7.49493271663381208413445258098, 8.730316433205199260591113950576, 9.515769365308114152556165995707, 10.66165116665396746950173295463

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