↑

Properties

Label 2-1216-152.45-c1-0-3
Degree $2$
Conductor $1216$
Sign $-0.837 - 0.547i$
Analytic cond. $9.70980$
Root an. cond. $3.11605$
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  + (0.495 − 0.285i)3-s + (−0.355 + 0.205i)5-s − 0.565·7-s + (−1.33 + 2.31i)9-s − 0.623i·11-s + (−3.87 − 2.23i)13-s + (−0.117 + 0.203i)15-s + (−1.03 − 1.79i)17-s + (1.77 + 3.98i)19-s + (−0.280 + 0.161i)21-s + (−2.39 + 4.15i)23-s + (−2.41 + 4.18i)25-s + 3.24i·27-s + (−2.28 − 1.31i)29-s − 5.12·31-s + ⋯
L(s)  = 1  + (0.285 − 0.165i)3-s + (−0.159 + 0.0918i)5-s − 0.213·7-s + (−0.445 + 0.771i)9-s − 0.187i·11-s + (−1.07 − 0.620i)13-s + (−0.0303 + 0.0525i)15-s + (−0.251 − 0.434i)17-s + (0.407 + 0.913i)19-s + (−0.0611 + 0.0352i)21-s + (−0.500 + 0.866i)23-s + (−0.483 + 0.836i)25-s + 0.624i·27-s + (−0.424 − 0.245i)29-s − 0.920·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.837 - 0.547i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.837 - 0.547i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1216\)    =    \(2^{6} \cdot 19\)
Sign: $-0.837 - 0.547i$
Analytic conductor: \(9.70980\)
Root analytic conductor: \(3.11605\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1216} (1185, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1216,\ (\ :1/2),\ -0.837 - 0.547i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4583801867\)
\(L(\frac12)\) \(\approx\) \(0.4583801867\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
19 \( 1 + (-1.77 - 3.98i)T \)
good3 \( 1 + (-0.495 + 0.285i)T + (1.5 - 2.59i)T^{2} \)
5 \( 1 + (0.355 - 0.205i)T + (2.5 - 4.33i)T^{2} \)
7 \( 1 + 0.565T + 7T^{2} \)
11 \( 1 + 0.623iT - 11T^{2} \)
13 \( 1 + (3.87 + 2.23i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (1.03 + 1.79i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (2.39 - 4.15i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (2.28 + 1.31i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + 5.12T + 31T^{2} \)
37 \( 1 + 5.19iT - 37T^{2} \)
41 \( 1 + (-1.11 - 1.93i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.626 + 0.361i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (4.18 - 7.24i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (10.5 + 6.07i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-7.71 + 4.45i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (0.0512 + 0.0295i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.447 - 0.258i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-4.32 - 7.49i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-2.86 - 4.95i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-4.50 - 7.79i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 10.2iT - 83T^{2} \)
89 \( 1 + (4.25 - 7.37i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (5.33 + 9.24i)T + (-48.5 + 84.0i)T^{2} \)
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

−9.782970343168855513525505062477, −9.489223153904656869170590317621, −8.153593807739948822809615250359, −7.76764868483808728474890220468, −6.96078768367280843799664325069, −5.67069878533802582527923105997, −5.17678244765481087839906982680, −3.82108243467324725908231702028, −2.90353462986698466671899579692, −1.82382325901516943993473226159, 0.17292206473704570440020272112, 2.07611692657607695660986941761, 3.13111895732558673730147901043, 4.16580105189817192240046783854, 4.98897294721528827336119354142, 6.19206926167663397755738654385, 6.85928341399823269588792141659, 7.81793694147073874012800145645, 8.712283432838530819782415964538, 9.390990061998500433831225362750

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