↑

Properties

Label 2-416-104.61-c1-0-11
Degree $2$
Conductor $416$
Sign $-0.966 - 0.256i$
Analytic cond. $3.32177$
Root an. cond. $1.82257$
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  + (−2.00 − 1.15i)3-s − 4.18i·5-s + (−0.818 − 1.41i)7-s + (1.17 + 2.04i)9-s + (−0.112 − 0.0649i)11-s + (−2.30 + 2.77i)13-s + (−4.83 + 8.38i)15-s + (1.60 + 2.77i)17-s + (3.31 − 1.91i)19-s + 3.78i·21-s + (1.09 − 1.89i)23-s − 12.4·25-s + 1.48i·27-s + (−4.98 − 2.87i)29-s − 4.16·31-s + ⋯
L(s)  = 1  + (−1.15 − 0.668i)3-s − 1.87i·5-s + (−0.309 − 0.535i)7-s + (0.392 + 0.680i)9-s + (−0.0339 − 0.0195i)11-s + (−0.640 + 0.768i)13-s + (−1.24 + 2.16i)15-s + (0.388 + 0.672i)17-s + (0.761 − 0.439i)19-s + 0.826i·21-s + (0.227 − 0.394i)23-s − 2.49·25-s + 0.286i·27-s + (−0.925 − 0.534i)29-s − 0.748·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.966 - 0.256i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.966 - 0.256i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(416\)    =    \(2^{5} \cdot 13\)
Sign: $-0.966 - 0.256i$
Analytic conductor: \(3.32177\)
Root analytic conductor: \(1.82257\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{416} (113, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 416,\ (\ :1/2),\ -0.966 - 0.256i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0697784 + 0.534467i\)
\(L(\frac12)\) \(\approx\) \(0.0697784 + 0.534467i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
13 \( 1 + (2.30 - 2.77i)T \)
good3 \( 1 + (2.00 + 1.15i)T + (1.5 + 2.59i)T^{2} \)
5 \( 1 + 4.18iT - 5T^{2} \)
7 \( 1 + (0.818 + 1.41i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (0.112 + 0.0649i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (-1.60 - 2.77i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3.31 + 1.91i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.09 + 1.89i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (4.98 + 2.87i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + 4.16T + 31T^{2} \)
37 \( 1 + (0.156 + 0.0902i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-2.25 + 3.90i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (5.84 - 3.37i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 - 4.71T + 47T^{2} \)
53 \( 1 - 4.75iT - 53T^{2} \)
59 \( 1 + (-9.37 + 5.40i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (1.24 - 0.719i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (11.0 + 6.36i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (2.53 + 4.39i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 - 3.28T + 73T^{2} \)
79 \( 1 - 9.44T + 79T^{2} \)
83 \( 1 - 5.48iT - 83T^{2} \)
89 \( 1 + (0.386 - 0.669i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (5.07 + 8.78i)T + (-48.5 + 84.0i)T^{2} \)
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

−10.95669122881547186061353174306, −9.707810363650379745581178139438, −8.990808433155076631858719772954, −7.82534239783299959478608117627, −6.92111208851324375110754908634, −5.76378457145035250237891201251, −5.07264069099332107774494745861, −4.03777637148770130665787872327, −1.60510664755144363975024433071, −0.40385308264980969263632116335, 2.66386265670147010860389934088, 3.63943135337966861634337997178, 5.29548498386717032865096225866, 5.84716705393627804674981448110, 6.95045651099996063178022416817, 7.67891292893791848278974381110, 9.447384737144750651270506495082, 10.14537102620490916307674487722, 10.76328057595729990851957751439, 11.53310700191051234122583199714

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