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Properties

Label 2-416-13.3-c1-0-3
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  + (1.20 + 2.09i)3-s + 2.82·5-s + (−2.20 + 3.82i)7-s + (−1.41 + 2.44i)9-s + (−1.62 − 2.80i)11-s + (−1 + 3.46i)13-s + (3.41 + 5.91i)15-s + (2.91 − 5.04i)17-s + (0.621 − 1.07i)19-s − 10.6·21-s + (−0.621 − 1.07i)23-s + 3.00·25-s + 0.414·27-s + (−4.32 − 7.49i)29-s + 5.65·31-s + ⋯
L(s)  = 1  + (0.696 + 1.20i)3-s + 1.26·5-s + (−0.834 + 1.44i)7-s + (−0.471 + 0.816i)9-s + (−0.488 − 0.846i)11-s + (−0.277 + 0.960i)13-s + (0.881 + 1.52i)15-s + (0.706 − 1.22i)17-s + (0.142 − 0.246i)19-s − 2.32·21-s + (−0.129 − 0.224i)23-s + 0.600·25-s + 0.0797·27-s + (−0.803 − 1.39i)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

\(L(1)\) \(\approx\) \(1.33304 + 1.31606i\)
\(L(\frac12)\) \(\approx\) \(1.33304 + 1.31606i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.20 - 2.09i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 - 2.82T + 5T^{2} \)
7 \( 1 + (2.20 - 3.82i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (1.62 + 2.80i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (-2.91 + 5.04i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.621 + 1.07i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.621 + 1.07i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (4.32 + 7.49i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 5.65T + 31T^{2} \)
37 \( 1 + (-3.74 - 6.48i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-2.91 - 5.04i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (2.03 - 3.52i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 6T + 47T^{2} \)
53 \( 1 + 2.82T + 53T^{2} \)
59 \( 1 + (-0.621 + 1.07i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (3.5 - 6.06i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (6.62 + 11.4i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-3.62 + 6.27i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 - 12.4T + 73T^{2} \)
79 \( 1 + 6T + 79T^{2} \)
83 \( 1 - 4T + 83T^{2} \)
89 \( 1 + (-1.67 - 2.89i)T + (-44.5 + 77.0i)T^{2} \)
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

−11.35484796692688337527970726096, −10.04944543107678862040808280419, −9.488616738015534926295207752071, −9.217563179927615778971584576186, −8.122432780260744564128465704578, −6.39630901918497497806049190205, −5.67336017977472620843637260124, −4.67384186769032496677940069073, −3.07031960212961596442441316760, −2.47081857615985642349475466340, 1.25426686852587588783194986356, 2.44918310669262376538627162911, 3.70892945981940961946842011871, 5.45303269820738617530664008474, 6.46960408175150434653360802389, 7.35168825284763161452944805537, 7.903609832611024544152735507094, 9.246555227633315378718886901213, 10.22384562239514511752953407474, 10.50893836827839743915766333553

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