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Properties

Label 2-1848-21.20-c1-0-15
Degree $2$
Conductor $1848$
Sign $-0.999 - 0.0235i$
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  + (−0.228 + 1.71i)3-s + 1.84·5-s + (0.286 + 2.63i)7-s + (−2.89 − 0.783i)9-s + i·11-s + 6.65i·13-s + (−0.420 + 3.16i)15-s − 6.91·17-s + 0.932i·19-s + (−4.58 − 0.107i)21-s − 6.36i·23-s − 1.60·25-s + (2.00 − 4.79i)27-s + 1.98i·29-s − 5.52i·31-s + ⋯
L(s)  = 1  + (−0.131 + 0.991i)3-s + 0.823·5-s + (0.108 + 0.994i)7-s + (−0.965 − 0.261i)9-s + 0.301i·11-s + 1.84i·13-s + (−0.108 + 0.816i)15-s − 1.67·17-s + 0.214i·19-s + (−0.999 − 0.0235i)21-s − 1.32i·23-s − 0.321·25-s + (0.386 − 0.922i)27-s + 0.367i·29-s − 0.991i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1848 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0235i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1848 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.0235i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1848\)    =    \(2^{3} \cdot 3 \cdot 7 \cdot 11\)
Sign: $-0.999 - 0.0235i$
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: \((2,\ 1848,\ (\ :1/2),\ -0.999 - 0.0235i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.197759362\)
\(L(\frac12)\) \(\approx\) \(1.197759362\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.228 - 1.71i)T \)
7 \( 1 + (-0.286 - 2.63i)T \)
11 \( 1 - iT \)
good5 \( 1 - 1.84T + 5T^{2} \)
13 \( 1 - 6.65iT - 13T^{2} \)
17 \( 1 + 6.91T + 17T^{2} \)
19 \( 1 - 0.932iT - 19T^{2} \)
23 \( 1 + 6.36iT - 23T^{2} \)
29 \( 1 - 1.98iT - 29T^{2} \)
31 \( 1 + 5.52iT - 31T^{2} \)
37 \( 1 - 0.344T + 37T^{2} \)
41 \( 1 + 1.92T + 41T^{2} \)
43 \( 1 + 1.37T + 43T^{2} \)
47 \( 1 - 10.2T + 47T^{2} \)
53 \( 1 - 6.02iT - 53T^{2} \)
59 \( 1 - 10.1T + 59T^{2} \)
61 \( 1 - 13.4iT - 61T^{2} \)
67 \( 1 + 7.35T + 67T^{2} \)
71 \( 1 - 7.58iT - 71T^{2} \)
73 \( 1 + 7.92iT - 73T^{2} \)
79 \( 1 + 0.993T + 79T^{2} \)
83 \( 1 + 6.57T + 83T^{2} \)
89 \( 1 + 9.35T + 89T^{2} \)
97 \( 1 + 11.9iT - 97T^{2} \)
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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

−9.539454332130853877277103127541, −8.885447997765118992072514297496, −8.619372938520836425752382901489, −7.04856824182550792369552715400, −6.24613293720421833230508335635, −5.67789720923659364802077889948, −4.55843351893707725847888857668, −4.18615590028924064330100986487, −2.57305020250774470865635500730, −2.01859100990961150643367073466, 0.41833773396557344057706124754, 1.57726200594017717597925053722, 2.63620649073121751979566028741, 3.66319340688328415898352771044, 5.04029118067748392007900177419, 5.70834784599859485855452827344, 6.52871228480435269468143164204, 7.24669447820703653449901601194, 7.977225146501749879378457467107, 8.692542353622288536586517129961

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