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Properties

Label 2-1848-21.20-c1-0-17
Degree $2$
Conductor $1848$
Sign $0.0594 - 0.998i$
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  + (1.35 − 1.08i)3-s + (−1.77 + 1.96i)7-s + (0.649 − 2.92i)9-s + i·11-s + 5.74i·13-s − 3.64·17-s − 0.413i·19-s + (−0.272 + 4.57i)21-s + 8.85i·23-s − 5·25-s + (−2.29 − 4.66i)27-s + 1.29i·29-s + 0.874i·31-s + (1.08 + 1.35i)33-s − 5.80·37-s + ⋯
L(s)  = 1  + (0.779 − 0.625i)3-s + (−0.671 + 0.741i)7-s + (0.216 − 0.976i)9-s + 0.301i·11-s + 1.59i·13-s − 0.884·17-s − 0.0949i·19-s + (−0.0594 + 0.998i)21-s + 1.84i·23-s − 25-s + (−0.442 − 0.896i)27-s + 0.241i·29-s + 0.157i·31-s + (0.188 + 0.235i)33-s − 0.954·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1848 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0594 - 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1848 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0594 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1848\)    =    \(2^{3} \cdot 3 \cdot 7 \cdot 11\)
Sign: $0.0594 - 0.998i$
Analytic conductor: \(14.7563\)
Root analytic conductor: \(3.84140\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1848} (881, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1848,\ (\ :1/2),\ 0.0594 - 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.382380395\)
\(L(\frac12)\) \(\approx\) \(1.382380395\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.35 + 1.08i)T \)
7 \( 1 + (1.77 - 1.96i)T \)
11 \( 1 - iT \)
good5 \( 1 + 5T^{2} \)
13 \( 1 - 5.74iT - 13T^{2} \)
17 \( 1 + 3.64T + 17T^{2} \)
19 \( 1 + 0.413iT - 19T^{2} \)
23 \( 1 - 8.85iT - 23T^{2} \)
29 \( 1 - 1.29iT - 29T^{2} \)
31 \( 1 - 0.874iT - 31T^{2} \)
37 \( 1 + 5.80T + 37T^{2} \)
41 \( 1 - 4.71T + 41T^{2} \)
43 \( 1 - 0.0536T + 43T^{2} \)
47 \( 1 - 2.70T + 47T^{2} \)
53 \( 1 - 10.7iT - 53T^{2} \)
59 \( 1 - 3.93T + 59T^{2} \)
61 \( 1 + 1.55iT - 61T^{2} \)
67 \( 1 + 11.1T + 67T^{2} \)
71 \( 1 + 8.56iT - 71T^{2} \)
73 \( 1 - 16.4iT - 73T^{2} \)
79 \( 1 - 12.0T + 79T^{2} \)
83 \( 1 - 9.82T + 83T^{2} \)
89 \( 1 - 3.78T + 89T^{2} \)
97 \( 1 - 8.52iT - 97T^{2} \)
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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

−9.207181862506235140691647897022, −8.911147536947799657916480347784, −7.80874628843499026588273871592, −7.05950782348621450541963334159, −6.44999588957516180685668585150, −5.58043700400703681827151660793, −4.28609506082178413542855691153, −3.48200900299961762923380928174, −2.37745367013322382650744510165, −1.64673433267187813634648027709, 0.43028832628210286867813097926, 2.27469885785908257439661444498, 3.17707724604882028002285561195, 3.94179468513236618408579224955, 4.78131895038128370750511021293, 5.82694518505036819689653106739, 6.73660350023985832301407088344, 7.69667404947951631575525724733, 8.285323617818591164814748115022, 9.057417503557948194532020931756

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