↑

Properties

Label 2-416-104.77-c1-0-11
Degree $2$
Conductor $416$
Sign $-0.937 - 0.347i$
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  − 2.94i·3-s − 1.81·5-s + 1.13i·7-s − 5.70·9-s − 4.40·11-s + (−2.58 − 2.50i)13-s + 5.35i·15-s + 0.701·17-s + 5.95·19-s + 3.36·21-s − 4·23-s − 1.70·25-s + 7.96i·27-s + 5.01i·29-s − 8.77i·31-s + ⋯
L(s)  = 1  − 1.70i·3-s − 0.812·5-s + 0.430i·7-s − 1.90·9-s − 1.32·11-s + (−0.717 − 0.696i)13-s + 1.38i·15-s + 0.170·17-s + 1.36·19-s + 0.733·21-s − 0.834·23-s − 0.340·25-s + 1.53i·27-s + 0.932i·29-s − 1.57i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.937 - 0.347i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.937 - 0.347i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(416\)    =    \(2^{5} \cdot 13\)
Sign: $-0.937 - 0.347i$
Analytic conductor: \(3.32177\)
Root analytic conductor: \(1.82257\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{416} (337, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 416,\ (\ :1/2),\ -0.937 - 0.347i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0915884 + 0.510208i\)
\(L(\frac12)\) \(\approx\) \(0.0915884 + 0.510208i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.58 + 2.50i)T \)
good3 \( 1 + 2.94iT - 3T^{2} \)
5 \( 1 + 1.81T + 5T^{2} \)
7 \( 1 - 1.13iT - 7T^{2} \)
11 \( 1 + 4.40T + 11T^{2} \)
17 \( 1 - 0.701T + 17T^{2} \)
19 \( 1 - 5.95T + 19T^{2} \)
23 \( 1 + 4T + 23T^{2} \)
29 \( 1 - 5.01iT - 29T^{2} \)
31 \( 1 + 8.77iT - 31T^{2} \)
37 \( 1 + 3.36T + 37T^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 + 2.94iT - 43T^{2} \)
47 \( 1 - 1.13iT - 47T^{2} \)
53 \( 1 + 11.7iT - 53T^{2} \)
59 \( 1 + 5.95T + 59T^{2} \)
61 \( 1 - 61T^{2} \)
67 \( 1 + 7.49T + 67T^{2} \)
71 \( 1 + 11.8iT - 71T^{2} \)
73 \( 1 + 8.43iT - 73T^{2} \)
79 \( 1 - 10.8T + 79T^{2} \)
83 \( 1 + 2.85T + 83T^{2} \)
89 \( 1 - 8.43iT - 89T^{2} \)
97 \( 1 + 12.9iT - 97T^{2} \)
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

−10.99178950585917456390087883236, −9.795732005704590118602089496325, −8.418973842480414703745405096928, −7.64176828531845958891868448990, −7.41213846336651088655333543357, −5.98815620957326193323621835520, −5.16052513303897782227303704355, −3.23351120880476578568853157210, −2.14780028193559796460141726380, −0.31544598020198545195161076507, 2.89520133806281409352524074924, 3.93996541154101015795917268034, 4.75462238730020055629205013888, 5.61105944559796117999197351782, 7.31300741327736845742177101069, 8.103132825183062001092502209776, 9.227934997177447201379396904028, 10.04883037980242445504051267279, 10.58957815903099972553793601997, 11.55493088385701449822211076504

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