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Properties

Label 2-1776-37.26-c1-0-25
Degree $2$
Conductor $1776$
Sign $-0.227 + 0.973i$
Analytic cond. $14.1814$
Root an. cond. $3.76582$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 + 0.866i)3-s + (−2 + 3.46i)5-s + (0.5 − 0.866i)7-s + (−0.499 − 0.866i)9-s + 2·11-s + (−1.5 + 2.59i)13-s + (−1.99 − 3.46i)15-s + (−4 − 6.92i)17-s + (−4 + 6.92i)19-s + (0.499 + 0.866i)21-s + (−5.49 − 9.52i)25-s + 0.999·27-s − 10·29-s + 7·31-s + (−1 + 1.73i)33-s + ⋯
L(s)  = 1  + (−0.288 + 0.499i)3-s + (−0.894 + 1.54i)5-s + (0.188 − 0.327i)7-s + (−0.166 − 0.288i)9-s + 0.603·11-s + (−0.416 + 0.720i)13-s + (−0.516 − 0.894i)15-s + (−0.970 − 1.68i)17-s + (−0.917 + 1.58i)19-s + (0.109 + 0.188i)21-s + (−1.09 − 1.90i)25-s + 0.192·27-s − 1.85·29-s + 1.25·31-s + (−0.174 + 0.301i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1776 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.227 + 0.973i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1776 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.227 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1776\)    =    \(2^{4} \cdot 3 \cdot 37\)
Sign: $-0.227 + 0.973i$
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: \(1\)
Selberg data: \((2,\ 1776,\ (\ :1/2),\ -0.227 + 0.973i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + (-5 - 3.46i)T \)
good5 \( 1 + (2 - 3.46i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (-0.5 + 0.866i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 - 2T + 11T^{2} \)
13 \( 1 + (1.5 - 2.59i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (4 + 6.92i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (4 - 6.92i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 + 10T + 29T^{2} \)
31 \( 1 - 7T + 31T^{2} \)
41 \( 1 + (-1 + 1.73i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + 5T + 43T^{2} \)
47 \( 1 - 6T + 47T^{2} \)
53 \( 1 + (3 + 5.19i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (6 + 10.3i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1 + 1.73i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-4.5 + 7.79i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-5 + 8.66i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + 11T + 73T^{2} \)
79 \( 1 + (4.5 - 7.79i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (6 + 10.3i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-3 - 5.19i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - T + 97T^{2} \)
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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

−9.254954390727516434006062771942, −8.118898558602027675216663693374, −7.36763639073063772048789327483, −6.71955138927057100818498158492, −6.07934058666188389188393230863, −4.66520700221105403108783765834, −4.05130963147098062814998672608, −3.23740954963212822670848682616, −2.12978183694363581366358963616, 0, 1.18964995933106277204164187028, 2.37118975271917916129188110272, 3.99439711479337234192795878666, 4.50715538721942426322936270007, 5.43922232891893558136014620058, 6.24761086752910059336088168915, 7.28109512434173496778651915468, 8.039462635349132737902234764260, 8.777498966348514908820641154364

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