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Properties

Label 2-416-416.411-c1-0-0
Degree $2$
Conductor $416$
Sign $-0.855 - 0.517i$
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  + (−1.37 + 0.323i)2-s + (−2.65 + 1.10i)3-s + (1.79 − 0.891i)4-s + (−1.41 − 3.42i)5-s + (3.30 − 2.37i)6-s − 0.353i·7-s + (−2.17 + 1.80i)8-s + (3.72 − 3.72i)9-s + (3.05 + 4.25i)10-s + (2.46 − 1.02i)11-s + (−3.77 + 4.33i)12-s + (−3.60 − 0.112i)13-s + (0.114 + 0.487i)14-s + (7.53 + 7.53i)15-s + (2.40 − 3.19i)16-s + 1.59·17-s + ⋯
L(s)  = 1  + (−0.973 + 0.229i)2-s + (−1.53 + 0.635i)3-s + (0.895 − 0.445i)4-s + (−0.633 − 1.52i)5-s + (1.34 − 0.969i)6-s − 0.133i·7-s + (−0.769 + 0.638i)8-s + (1.24 − 1.24i)9-s + (0.967 + 1.34i)10-s + (0.743 − 0.307i)11-s + (−1.08 + 1.25i)12-s + (−0.999 − 0.0311i)13-s + (0.0306 + 0.130i)14-s + (1.94 + 1.94i)15-s + (0.602 − 0.798i)16-s + 0.385·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.855 - 0.517i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.855 - 0.517i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(416\)    =    \(2^{5} \cdot 13\)
Sign: $-0.855 - 0.517i$
Analytic conductor: \(3.32177\)
Root analytic conductor: \(1.82257\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{416} (411, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 416,\ (\ :1/2),\ -0.855 - 0.517i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0148862 + 0.0533384i\)
\(L(\frac12)\) \(\approx\) \(0.0148862 + 0.0533384i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.37 - 0.323i)T \)
13 \( 1 + (3.60 + 0.112i)T \)
good3 \( 1 + (2.65 - 1.10i)T + (2.12 - 2.12i)T^{2} \)
5 \( 1 + (1.41 + 3.42i)T + (-3.53 + 3.53i)T^{2} \)
7 \( 1 + 0.353iT - 7T^{2} \)
11 \( 1 + (-2.46 + 1.02i)T + (7.77 - 7.77i)T^{2} \)
17 \( 1 - 1.59T + 17T^{2} \)
19 \( 1 + (1.92 - 4.64i)T + (-13.4 - 13.4i)T^{2} \)
23 \( 1 + (-2.45 + 2.45i)T - 23iT^{2} \)
29 \( 1 + (3.11 + 7.53i)T + (-20.5 + 20.5i)T^{2} \)
31 \( 1 + (5.89 - 5.89i)T - 31iT^{2} \)
37 \( 1 + (9.34 - 3.86i)T + (26.1 - 26.1i)T^{2} \)
41 \( 1 - 1.57T + 41T^{2} \)
43 \( 1 + (-0.399 + 0.963i)T + (-30.4 - 30.4i)T^{2} \)
47 \( 1 + (8.53 - 8.53i)T - 47iT^{2} \)
53 \( 1 + (1.84 - 4.45i)T + (-37.4 - 37.4i)T^{2} \)
59 \( 1 + (-0.791 - 1.91i)T + (-41.7 + 41.7i)T^{2} \)
61 \( 1 + (-2.66 - 6.43i)T + (-43.1 + 43.1i)T^{2} \)
67 \( 1 + (-14.2 - 5.91i)T + (47.3 + 47.3i)T^{2} \)
71 \( 1 + 8.29T + 71T^{2} \)
73 \( 1 - 2.92iT - 73T^{2} \)
79 \( 1 - 0.975T + 79T^{2} \)
83 \( 1 + (1.97 - 4.77i)T + (-58.6 - 58.6i)T^{2} \)
89 \( 1 - 5.86T + 89T^{2} \)
97 \( 1 + (0.254 + 0.254i)T + 97iT^{2} \)
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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

−11.57668300359699803015314066929, −10.61121305957155961869073524946, −9.795099657756404422798455196159, −8.989200511696063774964607758239, −8.041082991566829909741811066627, −6.91620948302486617255942410639, −5.78343941465127949972838230635, −5.06366826084770747288523065698, −4.00987694166509870636941749860, −1.22930531962712155180094636439, 0.06575726145176157326956976374, 2.03014390782833589874264269198, 3.53805747616077470603188365867, 5.32815495061723543150559491138, 6.67732419428803131758656311176, 6.96757112941024083691609415410, 7.63811056708660540096878567269, 9.206333863619544424360252087366, 10.27720964373777020509495588230, 11.01664006748404554611906070317

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