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Properties

Label 2-1776-37.26-c1-0-9
Degree $2$
Conductor $1776$
Sign $-0.418 - 0.908i$
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 + (−0.719 + 1.24i)5-s + (0.5 − 0.866i)7-s + (−0.499 − 0.866i)9-s + 0.950·11-s + (−0.219 + 0.380i)13-s + (−0.719 − 1.24i)15-s + (3.19 + 5.53i)17-s + (1.91 − 3.31i)19-s + (0.499 + 0.866i)21-s − 8.21·23-s + (1.46 + 2.53i)25-s + 0.999·27-s + 6.48·29-s + 3.38·31-s + ⋯
L(s)  = 1  + (−0.288 + 0.499i)3-s + (−0.321 + 0.557i)5-s + (0.188 − 0.327i)7-s + (−0.166 − 0.288i)9-s + 0.286·11-s + (−0.0609 + 0.105i)13-s + (−0.185 − 0.321i)15-s + (0.774 + 1.34i)17-s + (0.439 − 0.760i)19-s + (0.109 + 0.188i)21-s − 1.71·23-s + (0.292 + 0.507i)25-s + 0.192·27-s + 1.20·29-s + 0.608·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1776 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.418 - 0.908i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1776 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.418 - 0.908i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1776\)    =    \(2^{4} \cdot 3 \cdot 37\)
Sign: $-0.418 - 0.908i$
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: \((2,\ 1776,\ (\ :1/2),\ -0.418 - 0.908i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.199819538\)
\(L(\frac12)\) \(\approx\) \(1.199819538\)
\(L(\frac{3}{2})\) 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 \( 1 + (6.07 - 0.337i)T \)
good5 \( 1 + (0.719 - 1.24i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (-0.5 + 0.866i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 - 0.950T + 11T^{2} \)
13 \( 1 + (0.219 - 0.380i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-3.19 - 5.53i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.91 + 3.31i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + 8.21T + 23T^{2} \)
29 \( 1 - 6.48T + 29T^{2} \)
31 \( 1 - 3.38T + 31T^{2} \)
41 \( 1 + (3.86 - 6.69i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + 2.43T + 43T^{2} \)
47 \( 1 - 4.48T + 47T^{2} \)
53 \( 1 + (-1.63 - 2.83i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-0.964 - 1.67i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (0.0497 - 0.0861i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.46 + 4.26i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (7.87 - 13.6i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + 5.29T + 73T^{2} \)
79 \( 1 + (6.80 - 11.7i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (6.26 + 10.8i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-5.82 - 10.0i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 7.80T + 97T^{2} \)
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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

−9.776684893747392334048177552304, −8.676442151196465197312390589417, −8.023193374657991381280058485435, −7.11161067643028485482062880859, −6.34416563656452866152936566227, −5.52559042319112963373770776479, −4.46454684693792447709221053843, −3.77392280464903723109450931562, −2.83262113882848782735352237487, −1.31764411357861387455423650602, 0.50645748545470954220056869698, 1.74393277401759500875673489168, 2.95389193251140361262754187839, 4.10850655092381348243209885228, 5.07403749902751818861797293009, 5.72954423160524859287713755580, 6.67942843920167129238029912279, 7.52509828497159604846722877856, 8.241369828631822772938422104606, 8.839462667153414294991646531103

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