Properties

Label 2-416-13.3-c1-0-2
Degree $2$
Conductor $416$
Sign $0.0128 - 0.999i$
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  + (−0.207 − 0.358i)3-s − 2.82·5-s + (−0.792 + 1.37i)7-s + (1.41 − 2.44i)9-s + (2.62 + 4.54i)11-s + (−1 + 3.46i)13-s + (0.585 + 1.01i)15-s + (0.0857 − 0.148i)17-s + (−3.62 + 6.27i)19-s + 0.656·21-s + (3.62 + 6.27i)23-s + 3.00·25-s − 2.41·27-s + (1.32 + 2.30i)29-s − 5.65·31-s + ⋯
L(s)  = 1  + (−0.119 − 0.207i)3-s − 1.26·5-s + (−0.299 + 0.519i)7-s + (0.471 − 0.816i)9-s + (0.790 + 1.36i)11-s + (−0.277 + 0.960i)13-s + (0.151 + 0.261i)15-s + (0.0208 − 0.0360i)17-s + (−0.830 + 1.43i)19-s + 0.143·21-s + (0.755 + 1.30i)23-s + 0.600·25-s − 0.464·27-s + (0.246 + 0.427i)29-s − 1.01·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0128 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0128 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(416\)    =    \(2^{5} \cdot 13\)
Sign: $0.0128 - 0.999i$
Analytic conductor: \(3.32177\)
Root analytic conductor: \(1.82257\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{416} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 416,\ (\ :1/2),\ 0.0128 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.576056 + 0.568716i\)
\(L(\frac12)\) \(\approx\) \(0.576056 + 0.568716i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1 - 3.46i)T \)
good3 \( 1 + (0.207 + 0.358i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + 2.82T + 5T^{2} \)
7 \( 1 + (0.792 - 1.37i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-2.62 - 4.54i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (-0.0857 + 0.148i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.62 - 6.27i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3.62 - 6.27i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.32 - 2.30i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 5.65T + 31T^{2} \)
37 \( 1 + (4.74 + 8.21i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-0.0857 - 0.148i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-5.03 + 8.72i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 6T + 47T^{2} \)
53 \( 1 - 2.82T + 53T^{2} \)
59 \( 1 + (3.62 - 6.27i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (3.5 - 6.06i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.37 + 4.11i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (0.621 - 1.07i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + 4.48T + 73T^{2} \)
79 \( 1 + 6T + 79T^{2} \)
83 \( 1 - 4T + 83T^{2} \)
89 \( 1 + (-7.32 - 12.6i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-4.5 + 7.79i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.80393691613890849795877881789, −10.60069393281928365062442909483, −9.408868780057483205130325782576, −8.902690870134243469155965645063, −7.36033793133985790473398221516, −7.11120684372679080316926938719, −5.81696401799114879105412283180, −4.26256629918312557751062620821, −3.71599974571601652190490860191, −1.76323639653083310184307739574, 0.54647633518433370662244596702, 2.96603595367112199857029046969, 4.04840041830719252535537556835, 4.92692760236152258859330306472, 6.38828120616388474176376717509, 7.33462485768820063904887853516, 8.214888612049930276306394267408, 8.999573009015406530439808706843, 10.42982403381762458570429438831, 10.92929978484628411864442055441

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