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Properties

Label 2-416-104.99-c1-0-0
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: $\chi_{416} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 416,\ (\ :1/2),\ -0.957 - 0.289i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0516898 + 0.349093i\)
\(L(\frac12)\) \(\approx\) \(0.0516898 + 0.349093i\)
\(L(\frac{3}{2})\) 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

−11.40898402206881439816295049502, −10.78775943778824226801881685413, −9.894739706812234225542328830008, −8.865841785925640623721378297013, −7.71442207920456603963892291435, −7.14328859976291820371974682354, −6.29858617194998800922041737282, −4.44603574149262605827057381643, −3.48804977272946765309609836342, −2.67146770059946876187202292332, 0.19767752283746219961372471541, 2.65575089887556229422964292453, 3.57496313961042972605135272492, 4.99132920166081112136667617576, 5.81671496112109025206349593107, 7.37186392693435805117818798283, 8.183503863471162482817469421664, 8.976761544711955104843657128909, 9.465172401269568065799750120225, 10.96625780926382195448254554280

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