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Properties

Label 2-416-104.77-c1-0-7
Degree $2$
Conductor $416$
Sign $0.940 - 0.339i$
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  + 1.51i·3-s + 3.11·5-s − 2.77i·7-s + 0.701·9-s + 2.56·11-s + (−0.546 − 3.56i)13-s + 4.72i·15-s − 5.70·17-s + 4.75·19-s + 4.20·21-s − 4·23-s + 4.70·25-s + 5.61i·27-s + 7.12i·29-s + 3.60i·31-s + ⋯
L(s)  = 1  + 0.875i·3-s + 1.39·5-s − 1.04i·7-s + 0.233·9-s + 0.774·11-s + (−0.151 − 0.988i)13-s + 1.21i·15-s − 1.38·17-s + 1.09·19-s + 0.918·21-s − 0.834·23-s + 0.940·25-s + 1.07i·27-s + 1.32i·29-s + 0.647i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.940 - 0.339i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.940 - 0.339i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(416\)    =    \(2^{5} \cdot 13\)
Sign: $0.940 - 0.339i$
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: \((2,\ 416,\ (\ :1/2),\ 0.940 - 0.339i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.75875 + 0.307758i\)
\(L(\frac12)\) \(\approx\) \(1.75875 + 0.307758i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.546 + 3.56i)T \)
good3 \( 1 - 1.51iT - 3T^{2} \)
5 \( 1 - 3.11T + 5T^{2} \)
7 \( 1 + 2.77iT - 7T^{2} \)
11 \( 1 - 2.56T + 11T^{2} \)
17 \( 1 + 5.70T + 17T^{2} \)
19 \( 1 - 4.75T + 19T^{2} \)
23 \( 1 + 4T + 23T^{2} \)
29 \( 1 - 7.12iT - 29T^{2} \)
31 \( 1 - 3.60iT - 31T^{2} \)
37 \( 1 + 4.20T + 37T^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 - 1.51iT - 43T^{2} \)
47 \( 1 + 2.77iT - 47T^{2} \)
53 \( 1 - 6.06iT - 53T^{2} \)
59 \( 1 + 4.75T + 59T^{2} \)
61 \( 1 - 61T^{2} \)
67 \( 1 + 12.0T + 67T^{2} \)
71 \( 1 + 6.66iT - 71T^{2} \)
73 \( 1 + 14.9iT - 73T^{2} \)
79 \( 1 + 14.8T + 79T^{2} \)
83 \( 1 - 9.89T + 83T^{2} \)
89 \( 1 - 14.9iT - 89T^{2} \)
97 \( 1 + 3.89iT - 97T^{2} \)
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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

−10.71228027858858332203217299827, −10.44014896189404094984762441758, −9.557246379367141280519188011086, −8.929164385556903141952124531058, −7.41494115186487191755395528225, −6.50931201177228237391485183380, −5.37524612096859052668769657439, −4.43535038728920062679679394629, −3.27654894095302781052286603597, −1.53162463530100630702929313905, 1.71227923012217136605537555423, 2.37478966518320509513177758908, 4.36204051756860756457388019877, 5.76607165802356257623637503872, 6.36141387546324543441731183725, 7.21698060613627366231963889641, 8.579334779139182642535301433293, 9.387901269909905222414702035388, 9.959912867912744895748041288660, 11.48787497072350559332823188835

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