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Properties

Label 2-416-104.83-c1-0-6
Degree $2$
Conductor $416$
Sign $0.957 - 0.289i$
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  + 3-s + (2.54 + 2.54i)5-s + (2.54 − 2.54i)7-s − 2·9-s + (−1 − i)11-s + (2.54 + 2.54i)13-s + (2.54 + 2.54i)15-s − 3i·17-s + (−2 + 2i)19-s + (2.54 − 2.54i)21-s − 5.09·23-s + 7.99i·25-s − 5·27-s − 5.09i·29-s + (5.09 + 5.09i)31-s + ⋯
L(s)  = 1  + 0.577·3-s + (1.14 + 1.14i)5-s + (0.963 − 0.963i)7-s − 0.666·9-s + (−0.301 − 0.301i)11-s + (0.707 + 0.707i)13-s + (0.658 + 0.658i)15-s − 0.727i·17-s + (−0.458 + 0.458i)19-s + (0.556 − 0.556i)21-s − 1.06·23-s + 1.59i·25-s − 0.962·27-s − 0.946i·29-s + (0.915 + 0.915i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.957 - 0.289i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.957 - 0.289i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(416\)    =    \(2^{5} \cdot 13\)
Sign: $0.957 - 0.289i$
Analytic conductor: \(3.32177\)
Root analytic conductor: \(1.82257\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{416} (239, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 416,\ (\ :1/2),\ 0.957 - 0.289i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.98980 + 0.294628i\)
\(L(\frac12)\) \(\approx\) \(1.98980 + 0.294628i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.54 - 2.54i)T \)
good3 \( 1 - T + 3T^{2} \)
5 \( 1 + (-2.54 - 2.54i)T + 5iT^{2} \)
7 \( 1 + (-2.54 + 2.54i)T - 7iT^{2} \)
11 \( 1 + (1 + i)T + 11iT^{2} \)
17 \( 1 + 3iT - 17T^{2} \)
19 \( 1 + (2 - 2i)T - 19iT^{2} \)
23 \( 1 + 5.09T + 23T^{2} \)
29 \( 1 + 5.09iT - 29T^{2} \)
31 \( 1 + (-5.09 - 5.09i)T + 31iT^{2} \)
37 \( 1 + (2.54 - 2.54i)T - 37iT^{2} \)
41 \( 1 + (-6 + 6i)T - 41iT^{2} \)
43 \( 1 - iT - 43T^{2} \)
47 \( 1 + (-2.54 + 2.54i)T - 47iT^{2} \)
53 \( 1 - 5.09iT - 53T^{2} \)
59 \( 1 + (8 + 8i)T + 59iT^{2} \)
61 \( 1 - 61T^{2} \)
67 \( 1 + (3 - 3i)T - 67iT^{2} \)
71 \( 1 + (7.64 + 7.64i)T + 71iT^{2} \)
73 \( 1 + (6 + 6i)T + 73iT^{2} \)
79 \( 1 - 5.09iT - 79T^{2} \)
83 \( 1 + (5 - 5i)T - 83iT^{2} \)
89 \( 1 + (2 + 2i)T + 89iT^{2} \)
97 \( 1 + (7 - 7i)T - 97iT^{2} \)
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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

−10.98978815745216941247365647574, −10.47378333256809716956113449034, −9.522846882555152077004091591682, −8.460833272230017705349779398442, −7.60721889227404273118802203977, −6.53306096055565267878808075195, −5.68900522642042334314257742942, −4.21350629763050392852135877784, −2.93739711415506004510821247999, −1.83594166905495058642333256099, 1.64159201182266667141281234590, 2.63960168777552721901613107640, 4.44283662951956588683922956516, 5.56051991924676673029888604411, 5.98747868371912356386435455814, 7.949970144349250383813649320626, 8.546476828639421592298622594852, 9.075571284278865699110782216726, 10.11312287058664130854485052809, 11.17376927509350179159868371787

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