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Properties

Label 2-1872-156.155-c1-0-12
Degree $2$
Conductor $1872$
Sign $0.577 + 0.816i$
Analytic cond. $14.9479$
Root an. cond. $3.86626$
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  − 4.60·7-s + 4.24i·11-s + 3.60·13-s − 7.92i·17-s − 1.39·19-s − 5·25-s + 0.557i·29-s + 10.6·31-s − 12.7i·47-s + 14.2·49-s − 11.6i·53-s + 15.2i·59-s + 14.4·61-s + 7.39·67-s − 15.2i·71-s + ⋯
L(s)  = 1  − 1.74·7-s + 1.27i·11-s + 1.00·13-s − 1.92i·17-s − 0.319·19-s − 25-s + 0.103i·29-s + 1.90·31-s − 1.85i·47-s + 2.03·49-s − 1.59i·53-s + 1.99i·59-s + 1.84·61-s + 0.903·67-s − 1.81i·71-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.577 + 0.816i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.577 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1872\)    =    \(2^{4} \cdot 3^{2} \cdot 13\)
Sign: $0.577 + 0.816i$
Analytic conductor: \(14.9479\)
Root analytic conductor: \(3.86626\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1872} (1871, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1872,\ (\ :1/2),\ 0.577 + 0.816i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.141952310\)
\(L(\frac12)\) \(\approx\) \(1.141952310\)
\(L(\frac{3}{2})\) 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 \( 1 \)
13 \( 1 - 3.60T \)
good5 \( 1 + 5T^{2} \)
7 \( 1 + 4.60T + 7T^{2} \)
11 \( 1 - 4.24iT - 11T^{2} \)
17 \( 1 + 7.92iT - 17T^{2} \)
19 \( 1 + 1.39T + 19T^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 - 0.557iT - 29T^{2} \)
31 \( 1 - 10.6T + 31T^{2} \)
37 \( 1 - 37T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 - 43T^{2} \)
47 \( 1 + 12.7iT - 47T^{2} \)
53 \( 1 + 11.6iT - 53T^{2} \)
59 \( 1 - 15.2iT - 59T^{2} \)
61 \( 1 - 14.4T + 61T^{2} \)
67 \( 1 - 7.39T + 67T^{2} \)
71 \( 1 + 15.2iT - 71T^{2} \)
73 \( 1 - 73T^{2} \)
79 \( 1 - 79T^{2} \)
83 \( 1 + 15.2iT - 83T^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 - 97T^{2} \)
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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

−9.266193008570035345968496452469, −8.443310400553032069076851370191, −7.29053608588750217614767250286, −6.78914828638619981157443325315, −6.07358534850579545241896838197, −5.04394107415245849492733367332, −4.04586208434120495275714683878, −3.16735628306732704587525987823, −2.23616311900132125064109561252, −0.51162685152741464884809870403, 1.00650894235355430918405101569, 2.61981437153982871191208502993, 3.57771418308121765080808516747, 4.04667039713553171005608543203, 5.70066976497734717619291466699, 6.23623622381892379381570370398, 6.59700170377857069435110167580, 8.075104013619493071635906957814, 8.451402934180158092105527818867, 9.396924651760517818452513528004

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