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Properties

Label 2-416-104.101-c1-0-11
Degree $2$
Conductor $416$
Sign $-0.620 + 0.783i$
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  + (0.779 − 0.449i)3-s − 0.893·5-s + (−3.65 − 2.11i)7-s + (−1.09 + 1.89i)9-s + (−3.01 − 5.22i)11-s + (0.579 − 3.55i)13-s + (−0.696 + 0.401i)15-s + (1.00 − 1.74i)17-s + (1.59 − 2.75i)19-s − 3.79·21-s + (2.65 + 4.59i)23-s − 4.20·25-s + 4.67i·27-s + (4.50 − 2.60i)29-s + 2.51i·31-s + ⋯
L(s)  = 1  + (0.449 − 0.259i)3-s − 0.399·5-s + (−1.38 − 0.797i)7-s + (−0.365 + 0.632i)9-s + (−0.909 − 1.57i)11-s + (0.160 − 0.986i)13-s + (−0.179 + 0.103i)15-s + (0.243 − 0.422i)17-s + (0.365 − 0.632i)19-s − 0.828·21-s + (0.553 + 0.958i)23-s − 0.840·25-s + 0.898i·27-s + (0.836 − 0.482i)29-s + 0.451i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.620 + 0.783i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.620 + 0.783i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(416\)    =    \(2^{5} \cdot 13\)
Sign: $-0.620 + 0.783i$
Analytic conductor: \(3.32177\)
Root analytic conductor: \(1.82257\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{416} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 416,\ (\ :1/2),\ -0.620 + 0.783i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.354578 - 0.733283i\)
\(L(\frac12)\) \(\approx\) \(0.354578 - 0.733283i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.579 + 3.55i)T \)
good3 \( 1 + (-0.779 + 0.449i)T + (1.5 - 2.59i)T^{2} \)
5 \( 1 + 0.893T + 5T^{2} \)
7 \( 1 + (3.65 + 2.11i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (3.01 + 5.22i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (-1.00 + 1.74i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.59 + 2.75i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.65 - 4.59i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-4.50 + 2.60i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 - 2.51iT - 31T^{2} \)
37 \( 1 + (-1.00 - 1.73i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (0.0169 - 0.00981i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (3.45 + 1.99i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + 3.22iT - 47T^{2} \)
53 \( 1 + 3.34iT - 53T^{2} \)
59 \( 1 + (3.15 - 5.45i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (6.79 + 3.92i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.53 - 2.66i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (2.22 + 1.28i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + 12.1iT - 73T^{2} \)
79 \( 1 + 3.01T + 79T^{2} \)
83 \( 1 - 8.90T + 83T^{2} \)
89 \( 1 + (-7.52 + 4.34i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-9.79 - 5.65i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.78761653541608902147137883931, −10.11260405783182062798447658734, −8.964434716697229848924250013547, −7.998224460413094117934432235240, −7.39784126664568382834352843685, −6.14991723879539093768222123245, −5.16546772865901776016950273648, −3.37817411775867392814330440015, −2.98187787225891020165728441290, −0.47310308810520303004795712342, 2.37431332647631412317228241440, 3.43067652870412120143261809614, 4.55631771499019239213764270252, 5.97039710871862438747089688119, 6.80662101497242933302525199710, 7.935544145587798106863788915537, 9.006747113741043216269477264292, 9.632239618029485530169079011476, 10.34452261332002889854952725355, 11.79826956943794500771944001575

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