Properties

Label 2-616-7.2-c1-0-8
Degree $2$
Conductor $616$
Sign $0.605 - 0.795i$
Analytic cond. $4.91878$
Root an. cond. $2.21783$
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  + (0.5 + 0.866i)3-s + (−1 + 1.73i)5-s + (2.5 − 0.866i)7-s + (1 − 1.73i)9-s + (0.5 + 0.866i)11-s + 3·13-s − 1.99·15-s + (1 + 1.73i)17-s + (−2 + 3.46i)19-s + (2 + 1.73i)21-s + (2 − 3.46i)23-s + (0.500 + 0.866i)25-s + 5·27-s − 7·29-s + (4 + 6.92i)31-s + ⋯
L(s)  = 1  + (0.288 + 0.499i)3-s + (−0.447 + 0.774i)5-s + (0.944 − 0.327i)7-s + (0.333 − 0.577i)9-s + (0.150 + 0.261i)11-s + 0.832·13-s − 0.516·15-s + (0.242 + 0.420i)17-s + (−0.458 + 0.794i)19-s + (0.436 + 0.377i)21-s + (0.417 − 0.722i)23-s + (0.100 + 0.173i)25-s + 0.962·27-s − 1.29·29-s + (0.718 + 1.24i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 616 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 - 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 616 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.605 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(616\)    =    \(2^{3} \cdot 7 \cdot 11\)
Sign: $0.605 - 0.795i$
Analytic conductor: \(4.91878\)
Root analytic conductor: \(2.21783\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{616} (177, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 616,\ (\ :1/2),\ 0.605 - 0.795i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.57061 + 0.778596i\)
\(L(\frac12)\) \(\approx\) \(1.57061 + 0.778596i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.5 + 0.866i)T \)
11 \( 1 + (-0.5 - 0.866i)T \)
good3 \( 1 + (-0.5 - 0.866i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (1 - 1.73i)T + (-2.5 - 4.33i)T^{2} \)
13 \( 1 - 3T + 13T^{2} \)
17 \( 1 + (-1 - 1.73i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2 - 3.46i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2 + 3.46i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 7T + 29T^{2} \)
31 \( 1 + (-4 - 6.92i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (6 - 10.3i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 8T + 41T^{2} \)
43 \( 1 - 8T + 43T^{2} \)
47 \( 1 + (-5 + 8.66i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (7 + 12.1i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-4.5 - 7.79i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.5 + 4.33i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.5 - 2.59i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 6T + 71T^{2} \)
73 \( 1 + (2 + 3.46i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-8.5 + 14.7i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 6T + 83T^{2} \)
89 \( 1 + (-1 + 1.73i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 7T + 97T^{2} \)
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

−10.54846231530291366216059357486, −10.20451951956080376780096031976, −8.858057339557577074009056020645, −8.265289383306845538980080572280, −7.16880092155352653107387323296, −6.44289038799440067070313190861, −5.05442668393773005655721811827, −3.97481418847308627422465830391, −3.31492694623035432685268894190, −1.55545249153361650259548400912, 1.13869421553439109130755674451, 2.36834247783831320477779412616, 3.98959665890938924282397788452, 4.90653618310098536125358877522, 5.84610373072473398863765096577, 7.24166005236254877792288094451, 7.890289421941768626880380601976, 8.666161162301102288123955656529, 9.309461789239942867867582369687, 10.80628437902855861170270112414

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