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Properties

Label 2-1872-13.9-c1-0-21
Degree $2$
Conductor $1872$
Sign $0.996 + 0.0891i$
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  + 3.17·5-s + (−1.08 − 1.88i)7-s + (1.45 − 2.52i)11-s + (−1.21 + 3.39i)13-s + (3.04 + 5.27i)17-s + (1.45 + 2.52i)19-s + (0.281 − 0.488i)23-s + 5.09·25-s + (−1.32 + 2.30i)29-s + 7.09·31-s + (−3.45 − 5.99i)35-s + (3.76 − 6.52i)37-s + (1.30 − 2.26i)41-s + (5.00 + 8.67i)43-s − 3.43·47-s + ⋯
L(s)  = 1  + 1.42·5-s + (−0.411 − 0.712i)7-s + (0.439 − 0.762i)11-s + (−0.337 + 0.941i)13-s + (0.739 + 1.28i)17-s + (0.334 + 0.579i)19-s + (0.0587 − 0.101i)23-s + 1.01·25-s + (−0.246 + 0.427i)29-s + 1.27·31-s + (−0.584 − 1.01i)35-s + (0.619 − 1.07i)37-s + (0.204 − 0.353i)41-s + (0.763 + 1.32i)43-s − 0.501·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 + 0.0891i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.996 + 0.0891i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1872\)    =    \(2^{4} \cdot 3^{2} \cdot 13\)
Sign: $0.996 + 0.0891i$
Analytic conductor: \(14.9479\)
Root analytic conductor: \(3.86626\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1872} (1153, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1872,\ (\ :1/2),\ 0.996 + 0.0891i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.317185866\)
\(L(\frac12)\) \(\approx\) \(2.317185866\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.21 - 3.39i)T \)
good5 \( 1 - 3.17T + 5T^{2} \)
7 \( 1 + (1.08 + 1.88i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-1.45 + 2.52i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (-3.04 - 5.27i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.45 - 2.52i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.281 + 0.488i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.32 - 2.30i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 7.09T + 31T^{2} \)
37 \( 1 + (-3.76 + 6.52i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-1.30 + 2.26i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-5.00 - 8.67i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 3.43T + 47T^{2} \)
53 \( 1 - 7.17T + 53T^{2} \)
59 \( 1 + (7.27 + 12.5i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (4.39 + 7.61i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-5.52 + 9.56i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-3.71 - 6.44i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 - 13.1T + 73T^{2} \)
79 \( 1 + 9.96T + 79T^{2} \)
83 \( 1 + 13.7T + 83T^{2} \)
89 \( 1 + (-2 + 3.46i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (0.629 + 1.09i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.501079001348323357642421378672, −8.521289670389591962655663625517, −7.64954390613504262503805022574, −6.54053416603374117647541684846, −6.18403118209542199324632557636, −5.35151622779645485891439279930, −4.19108615661690002286643051246, −3.33765821272031546562398459790, −2.09495089143631901305287709461, −1.12307115204002948861652731980, 1.08836917233323251309414422807, 2.47546386411313760821907589366, 2.91114420933049905807591559783, 4.50184331814693491149872157816, 5.41438403752947480455988337364, 5.87735717623353622550117208357, 6.81698214611432462014871571593, 7.56122382369590400356803298639, 8.665963229186331890622353480982, 9.513310604971127117218912424675

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