Properties

Label 2-416-13.12-c1-0-2
Degree $2$
Conductor $416$
Sign $-i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + 4.60i·7-s − 3·9-s + 0.605i·11-s + 3.60·13-s − 7.21·17-s + 8.60i·19-s + 5·25-s + 7.21·29-s − 3.39i·31-s + 12.6i·47-s − 14.2·49-s − 2·53-s − 9.81i·59-s + 6·61-s − 13.8i·63-s + ⋯
L(s)  = 1  + 1.74i·7-s − 9-s + 0.182i·11-s + 1.00·13-s − 1.74·17-s + 1.97i·19-s + 25-s + 1.33·29-s − 0.609i·31-s + 1.83i·47-s − 2.03·49-s − 0.274·53-s − 1.27i·59-s + 0.768·61-s − 1.74i·63-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(416\)    =    \(2^{5} \cdot 13\)
Sign: $-i$
Analytic conductor: \(3.32177\)
Root analytic conductor: \(1.82257\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{416} (129, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 416,\ (\ :1/2),\ -i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.796261 + 0.796261i\)
\(L(\frac12)\) \(\approx\) \(0.796261 + 0.796261i\)
\(L(\frac{3}{2})\) 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 \( 1 - 3.60T \)
good3 \( 1 + 3T^{2} \)
5 \( 1 - 5T^{2} \)
7 \( 1 - 4.60iT - 7T^{2} \)
11 \( 1 - 0.605iT - 11T^{2} \)
17 \( 1 + 7.21T + 17T^{2} \)
19 \( 1 - 8.60iT - 19T^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 - 7.21T + 29T^{2} \)
31 \( 1 + 3.39iT - 31T^{2} \)
37 \( 1 - 37T^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 + 43T^{2} \)
47 \( 1 - 12.6iT - 47T^{2} \)
53 \( 1 + 2T + 53T^{2} \)
59 \( 1 + 9.81iT - 59T^{2} \)
61 \( 1 - 6T + 61T^{2} \)
67 \( 1 + 7.39iT - 67T^{2} \)
71 \( 1 + 5.81iT - 71T^{2} \)
73 \( 1 - 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 + 17.8iT - 83T^{2} \)
89 \( 1 - 89T^{2} \)
97 \( 1 - 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.48308413684132509939852473486, −10.70945337469870713271168118268, −9.390086620278790879119792793450, −8.649255121889692142486832193404, −8.139682816425469971211075865347, −6.34486026994241942371625440363, −5.93834191756031048581914741482, −4.73265220071857005222764546477, −3.18671270363421497151561504739, −2.06751429733452498427171523657, 0.72874106844732925400529890387, 2.79257035189951538222292282755, 4.05690055713297330648870858632, 5.00004897850268835114046377351, 6.58919352020252607552271392207, 6.98116411676419319043567600418, 8.446031963934661764694942469765, 8.930508368986412365801052672009, 10.36959634230983350571206324619, 10.98826745456527218978602120433

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