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Properties

Label 2-416-104.11-c1-0-5
Degree $2$
Conductor $416$
Sign $0.994 + 0.105i$
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  + (0.223 + 0.386i)3-s + (0.612 − 0.612i)5-s + (−2.00 + 0.537i)7-s + (1.40 − 2.42i)9-s + (3.04 + 0.816i)11-s + (3.45 − 1.02i)13-s + (0.373 + 0.100i)15-s + (2.68 + 1.55i)17-s + (3.53 − 0.948i)19-s + (−0.655 − 0.655i)21-s + (−1.56 − 2.70i)23-s + 4.24i·25-s + 2.58·27-s + (−6.13 + 3.54i)29-s + (2.77 − 2.77i)31-s + ⋯
L(s)  = 1  + (0.128 + 0.223i)3-s + (0.274 − 0.274i)5-s + (−0.758 + 0.203i)7-s + (0.466 − 0.808i)9-s + (0.918 + 0.246i)11-s + (0.958 − 0.283i)13-s + (0.0964 + 0.0258i)15-s + (0.651 + 0.376i)17-s + (0.812 − 0.217i)19-s + (−0.143 − 0.143i)21-s + (−0.325 − 0.563i)23-s + 0.849i·25-s + 0.497·27-s + (−1.13 + 0.657i)29-s + (0.498 − 0.498i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 + 0.105i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.994 + 0.105i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(416\)    =    \(2^{5} \cdot 13\)
Sign: $0.994 + 0.105i$
Analytic conductor: \(3.32177\)
Root analytic conductor: \(1.82257\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{416} (271, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 416,\ (\ :1/2),\ 0.994 + 0.105i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.55949 - 0.0824995i\)
\(L(\frac12)\) \(\approx\) \(1.55949 - 0.0824995i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.45 + 1.02i)T \)
good3 \( 1 + (-0.223 - 0.386i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (-0.612 + 0.612i)T - 5iT^{2} \)
7 \( 1 + (2.00 - 0.537i)T + (6.06 - 3.5i)T^{2} \)
11 \( 1 + (-3.04 - 0.816i)T + (9.52 + 5.5i)T^{2} \)
17 \( 1 + (-2.68 - 1.55i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3.53 + 0.948i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + (1.56 + 2.70i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (6.13 - 3.54i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-2.77 + 2.77i)T - 31iT^{2} \)
37 \( 1 + (-0.155 + 0.580i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + (-1.60 + 5.98i)T + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (5.34 + 3.08i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-6.19 - 6.19i)T + 47iT^{2} \)
53 \( 1 - 2.19iT - 53T^{2} \)
59 \( 1 + (1.67 + 6.25i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (0.346 + 0.199i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.40 - 8.98i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + (4.33 + 16.1i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 + (6.53 - 6.53i)T - 73iT^{2} \)
79 \( 1 - 12.9iT - 79T^{2} \)
83 \( 1 + (9.22 + 9.22i)T + 83iT^{2} \)
89 \( 1 + (6.08 + 1.62i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (15.1 - 4.07i)T + (84.0 - 48.5i)T^{2} \)
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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

−11.20138976877147838647232567080, −10.06818684946055567286923219240, −9.374093946266642039551185602901, −8.772471204969461533938746657278, −7.39413152370800867199740031604, −6.40001467263431638089263736769, −5.58971591963141337463112436975, −4.06180393162180456770330781669, −3.25969224495653818887160747231, −1.31142776228596571101040923678, 1.48057434249162249220499169752, 3.09494249086049786186143770143, 4.17475719301177082951777794689, 5.66206827886868423152150925094, 6.54923057110409687760166376281, 7.44332519243945192643112913733, 8.459253951447021352369477772863, 9.595259332858188807232799748378, 10.15912819575778858352718355918, 11.27654332533081537204065027802

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