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Properties

Label 2-1848-7.4-c1-0-21
Degree $2$
Conductor $1848$
Sign $0.319 + 0.947i$
Analytic cond. $14.7563$
Root an. cond. $3.84140$
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 + (0.751 + 1.30i)5-s + (−2.44 + 1.00i)7-s + (−0.499 − 0.866i)9-s + (−0.5 + 0.866i)11-s − 3.15·13-s − 1.50·15-s + (0.761 − 1.31i)17-s + (−2.84 − 4.92i)19-s + (0.356 − 2.62i)21-s + (−2.70 − 4.67i)23-s + (1.36 − 2.37i)25-s + 0.999·27-s − 0.972·29-s + (1.57 − 2.73i)31-s + ⋯
L(s)  = 1  + (−0.288 + 0.499i)3-s + (0.336 + 0.582i)5-s + (−0.925 + 0.378i)7-s + (−0.166 − 0.288i)9-s + (−0.150 + 0.261i)11-s − 0.875·13-s − 0.388·15-s + (0.184 − 0.320i)17-s + (−0.652 − 1.13i)19-s + (0.0777 − 0.572i)21-s + (−0.563 − 0.975i)23-s + (0.273 − 0.474i)25-s + 0.192·27-s − 0.180·29-s + (0.283 − 0.491i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1848 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.319 + 0.947i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1848 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.319 + 0.947i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1848\)    =    \(2^{3} \cdot 3 \cdot 7 \cdot 11\)
Sign: $0.319 + 0.947i$
Analytic conductor: \(14.7563\)
Root analytic conductor: \(3.84140\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1848} (529, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1848,\ (\ :1/2),\ 0.319 + 0.947i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6498225669\)
\(L(\frac12)\) \(\approx\) \(0.6498225669\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.5 - 0.866i)T \)
7 \( 1 + (2.44 - 1.00i)T \)
11 \( 1 + (0.5 - 0.866i)T \)
good5 \( 1 + (-0.751 - 1.30i)T + (-2.5 + 4.33i)T^{2} \)
13 \( 1 + 3.15T + 13T^{2} \)
17 \( 1 + (-0.761 + 1.31i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.84 + 4.92i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (2.70 + 4.67i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 0.972T + 29T^{2} \)
31 \( 1 + (-1.57 + 2.73i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-3.40 - 5.90i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 10.1T + 41T^{2} \)
43 \( 1 + 4.60T + 43T^{2} \)
47 \( 1 + (-3.58 - 6.20i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-2.40 + 4.16i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-3.35 + 5.80i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (5.64 + 9.78i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.58 + 2.74i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 14.9T + 71T^{2} \)
73 \( 1 + (3.74 - 6.48i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-2.07 - 3.59i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 6.21T + 83T^{2} \)
89 \( 1 + (2.34 + 4.06i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 16.2T + 97T^{2} \)
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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

−9.340197201645978157441539985888, −8.433640071750200726405864511205, −7.35622724039550201469108137723, −6.53553023729140982180012071012, −6.05434462368530895715962523848, −4.94969461154756680519973484007, −4.23016892965427194875540083614, −2.90001092208094026913255093135, −2.42362082308313400676037813854, −0.26227808456182913120575017068, 1.17075675309885809318640722983, 2.35721433617301027585614717032, 3.54087332686248179182044249008, 4.48834054843219750570732406209, 5.71735758036934815086756007184, 5.96070020111992574930636116737, 7.16215989142872149335342446028, 7.63746317401603196753078612559, 8.676302206505829683053947931168, 9.400039265993406267233546500414

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