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Properties

Label 2-1776-37.26-c1-0-29
Degree $2$
Conductor $1776$
Sign $0.556 + 0.830i$
Analytic cond. $14.1814$
Root an. cond. $3.76582$
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  + (−0.5 + 0.866i)3-s + (2.05 − 3.55i)5-s + (0.5 − 0.866i)7-s + (−0.499 − 0.866i)9-s + 4.61·11-s + (2.55 − 4.41i)13-s + (2.05 + 3.55i)15-s + (2.25 + 3.90i)17-s + (−1.79 + 3.10i)19-s + (0.499 + 0.866i)21-s + 1.07·23-s + (−5.90 − 10.2i)25-s + 0.999·27-s − 2.71·29-s + 1.51·31-s + ⋯
L(s)  = 1  + (−0.288 + 0.499i)3-s + (0.916 − 1.58i)5-s + (0.188 − 0.327i)7-s + (−0.166 − 0.288i)9-s + 1.39·11-s + (0.707 − 1.22i)13-s + (0.529 + 0.916i)15-s + (0.547 + 0.947i)17-s + (−0.411 + 0.712i)19-s + (0.109 + 0.188i)21-s + 0.224·23-s + (−1.18 − 2.04i)25-s + 0.192·27-s − 0.503·29-s + 0.271·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1776 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.556 + 0.830i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1776 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.556 + 0.830i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1776\)    =    \(2^{4} \cdot 3 \cdot 37\)
Sign: $0.556 + 0.830i$
Analytic conductor: \(14.1814\)
Root analytic conductor: \(3.76582\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1776} (433, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1776,\ (\ :1/2),\ 0.556 + 0.830i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.126777986\)
\(L(\frac12)\) \(\approx\) \(2.126777986\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (0.5 - 0.866i)T \)
37 \( 1 + (-5.94 + 1.28i)T \)
good5 \( 1 + (-2.05 + 3.55i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (-0.5 + 0.866i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 - 4.61T + 11T^{2} \)
13 \( 1 + (-2.55 + 4.41i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-2.25 - 3.90i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.79 - 3.10i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 - 1.07T + 23T^{2} \)
29 \( 1 + 2.71T + 29T^{2} \)
31 \( 1 - 1.51T + 31T^{2} \)
41 \( 1 + (3.81 - 6.61i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 - 3.10T + 43T^{2} \)
47 \( 1 + 4.71T + 47T^{2} \)
53 \( 1 + (4.84 + 8.39i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (6.40 + 11.0i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-3.61 + 6.25i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (4.90 - 8.49i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-3.20 + 5.54i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + 10.7T + 73T^{2} \)
79 \( 1 + (1.21 - 2.10i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-6.68 - 11.5i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (1.58 + 2.75i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 18.0T + 97T^{2} \)
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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

−9.281832544716696409411881806043, −8.420928703164722867773417843156, −7.964395295064561465412350708967, −6.32930827917140836345199946566, −5.94550313560302016967051316954, −5.09554855153066811767848549543, −4.27403466098081506761784562134, −3.48565982950266587089015244393, −1.67455491053641453097042128252, −0.946325421377048562177983274601, 1.44377965240442175985326601489, 2.34772121556820868462752969001, 3.30607549029337063682697434853, 4.44261027043228528982157763368, 5.74906358315810209725139473226, 6.31512996543727198006745923446, 6.90413450829881270908402609847, 7.46314590726743484454068350293, 8.920383199138195133587417044134, 9.287908084768961401141541576306

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