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Properties

Label 2-1776-37.26-c1-0-21
Degree $2$
Conductor $1776$
Sign $0.907 + 0.421i$
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 + (−1.64 + 2.85i)5-s + (−0.00433 + 0.00749i)7-s + (−0.499 − 0.866i)9-s − 1.16·11-s + (2.14 − 3.72i)13-s + (1.64 + 2.85i)15-s + (−2.23 − 3.86i)17-s + (0.229 − 0.397i)19-s + (0.00433 + 0.00749i)21-s + 0.402·23-s + (−2.93 − 5.08i)25-s − 0.999·27-s + 7.17·29-s + 5.83·31-s + ⋯
L(s)  = 1  + (0.288 − 0.499i)3-s + (−0.737 + 1.27i)5-s + (−0.00163 + 0.00283i)7-s + (−0.166 − 0.288i)9-s − 0.351·11-s + (0.595 − 1.03i)13-s + (0.425 + 0.737i)15-s + (−0.541 − 0.937i)17-s + (0.0526 − 0.0911i)19-s + (0.000944 + 0.00163i)21-s + 0.0840·23-s + (−0.586 − 1.01i)25-s − 0.192·27-s + 1.33·29-s + 1.04·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1776 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.907 + 0.421i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1776 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.907 + 0.421i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1776\)    =    \(2^{4} \cdot 3 \cdot 37\)
Sign: $0.907 + 0.421i$
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: \((2,\ 1776,\ (\ :1/2),\ 0.907 + 0.421i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.564420021\)
\(L(\frac12)\) \(\approx\) \(1.564420021\)
\(L(\frac{3}{2})\) 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 \( 1 + (-4.41 + 4.18i)T \)
good5 \( 1 + (1.64 - 2.85i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (0.00433 - 0.00749i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + 1.16T + 11T^{2} \)
13 \( 1 + (-2.14 + 3.72i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (2.23 + 3.86i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.229 + 0.397i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 - 0.402T + 23T^{2} \)
29 \( 1 - 7.17T + 29T^{2} \)
31 \( 1 - 5.83T + 31T^{2} \)
41 \( 1 + (6.38 - 11.0i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 - 6.62T + 43T^{2} \)
47 \( 1 - 8.53T + 47T^{2} \)
53 \( 1 + (4.73 + 8.20i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-3.87 - 6.71i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.62 + 2.81i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-7.35 + 12.7i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-5.64 + 9.77i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 - 12.1T + 73T^{2} \)
79 \( 1 + (7.39 - 12.8i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-3.51 - 6.08i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (0.0164 + 0.0284i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 2.87T + 97T^{2} \)
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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

−9.184111686278143975416773221199, −8.035865264871472068523323162456, −7.86183515244440267965537847372, −6.79554663920328340912562073610, −6.38488285494204527567027521348, −5.19028098541347028456891720100, −4.05321965691939034960233671680, −2.99964285329605467016756249878, −2.59029888003377678814846001534, −0.74282228842898087034376786112, 0.983074660389323369134112256159, 2.34484660225398421802243538792, 3.77235389994983912460838538552, 4.30495779350777906078355193625, 5.02192146847930581130959249161, 6.05091777066039805454078772386, 7.03771182297083118108017987081, 8.150330313847379246241775926561, 8.561484518128616543269042507086, 9.087980989832612486950610413567

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