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Properties

Label 2-416-13.7-c0-0-0
Degree $2$
Conductor $416$
Sign $0.958 - 0.283i$
Analytic cond. $0.207611$
Root an. cond. $0.455643$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.366 + 0.366i)5-s + (0.5 + 0.866i)9-s + (−0.866 − 0.5i)13-s + (0.866 − 0.5i)17-s − 0.732i·25-s + (−0.866 + 1.5i)29-s + (−0.5 − 1.86i)37-s + (−1.86 + 0.5i)41-s + (−0.133 + 0.5i)45-s + (−0.866 − 0.5i)49-s + 53-s + (−0.5 − 0.866i)61-s + (−0.133 − 0.5i)65-s + (−1.36 + 1.36i)73-s + (−0.499 + 0.866i)81-s + ⋯
L(s)  = 1  + (0.366 + 0.366i)5-s + (0.5 + 0.866i)9-s + (−0.866 − 0.5i)13-s + (0.866 − 0.5i)17-s − 0.732i·25-s + (−0.866 + 1.5i)29-s + (−0.5 − 1.86i)37-s + (−1.86 + 0.5i)41-s + (−0.133 + 0.5i)45-s + (−0.866 − 0.5i)49-s + 53-s + (−0.5 − 0.866i)61-s + (−0.133 − 0.5i)65-s + (−1.36 + 1.36i)73-s + (−0.499 + 0.866i)81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.958 - 0.283i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.958 - 0.283i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(416\)    =    \(2^{5} \cdot 13\)
Sign: $0.958 - 0.283i$
Analytic conductor: \(0.207611\)
Root analytic conductor: \(0.455643\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{416} (33, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 416,\ (\ :0),\ 0.958 - 0.283i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8911503989\)
\(L(\frac12)\) \(\approx\) \(0.8911503989\)
\(L(1)\) 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 \( 1 + (0.866 + 0.5i)T \)
good3 \( 1 + (-0.5 - 0.866i)T^{2} \)
5 \( 1 + (-0.366 - 0.366i)T + iT^{2} \)
7 \( 1 + (0.866 + 0.5i)T^{2} \)
11 \( 1 + (-0.866 + 0.5i)T^{2} \)
17 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
19 \( 1 + (-0.866 - 0.5i)T^{2} \)
23 \( 1 + (0.5 + 0.866i)T^{2} \)
29 \( 1 + (0.866 - 1.5i)T + (-0.5 - 0.866i)T^{2} \)
31 \( 1 + iT^{2} \)
37 \( 1 + (0.5 + 1.86i)T + (-0.866 + 0.5i)T^{2} \)
41 \( 1 + (1.86 - 0.5i)T + (0.866 - 0.5i)T^{2} \)
43 \( 1 + (0.5 - 0.866i)T^{2} \)
47 \( 1 - iT^{2} \)
53 \( 1 - T + T^{2} \)
59 \( 1 + (0.866 + 0.5i)T^{2} \)
61 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (0.866 - 0.5i)T^{2} \)
71 \( 1 + (-0.866 - 0.5i)T^{2} \)
73 \( 1 + (1.36 - 1.36i)T - iT^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 + iT^{2} \)
89 \( 1 + (0.366 + 1.36i)T + (-0.866 + 0.5i)T^{2} \)
97 \( 1 + (-0.366 + 1.36i)T + (-0.866 - 0.5i)T^{2} \)
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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

−11.39754579220163190747623977517, −10.35597265870804675190652199141, −9.961280188652585584949896535698, −8.750886852822545663442337388415, −7.61326809478117963804206803296, −7.00011834840051689334266350962, −5.61215388254288460131656662988, −4.82904673324155750816642746617, −3.30254279028667025297886471347, −2.00582147086171783241045132741, 1.66165399736803874779821910051, 3.36312460119762353447156180102, 4.55797258030583618024191188667, 5.67083369729254006592423517546, 6.69863763700393266575282839627, 7.65580079574047187369217780307, 8.789294415773998369384167854444, 9.697359790074544399240236655573, 10.19130708723767703346990994858, 11.64258655047382734464132916831

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