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Properties

Label 2-1872-13.9-c1-0-17
Degree $2$
Conductor $1872$
Sign $0.0128 + 0.999i$
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  − 3·5-s + (1 + 1.73i)7-s + (−3 + 5.19i)11-s + (−3.5 − 0.866i)13-s + (−1.5 − 2.59i)17-s + (1 + 1.73i)19-s + (3 − 5.19i)23-s + 4·25-s + (1.5 − 2.59i)29-s + 4·31-s + (−3 − 5.19i)35-s + (3.5 − 6.06i)37-s + (−1.5 + 2.59i)41-s + (−5 − 8.66i)43-s + 6·47-s + ⋯
L(s)  = 1  − 1.34·5-s + (0.377 + 0.654i)7-s + (−0.904 + 1.56i)11-s + (−0.970 − 0.240i)13-s + (−0.363 − 0.630i)17-s + (0.229 + 0.397i)19-s + (0.625 − 1.08i)23-s + 0.800·25-s + (0.278 − 0.482i)29-s + 0.718·31-s + (−0.507 − 0.878i)35-s + (0.575 − 0.996i)37-s + (−0.234 + 0.405i)41-s + (−0.762 − 1.32i)43-s + 0.875·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0128 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0128 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1872\)    =    \(2^{4} \cdot 3^{2} \cdot 13\)
Sign: $0.0128 + 0.999i$
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: \((2,\ 1872,\ (\ :1/2),\ 0.0128 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5443233238\)
\(L(\frac12)\) \(\approx\) \(0.5443233238\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.5 + 0.866i)T \)
good5 \( 1 + 3T + 5T^{2} \)
7 \( 1 + (-1 - 1.73i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (3 - 5.19i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (1.5 + 2.59i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1 - 1.73i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-3 + 5.19i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.5 + 2.59i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 4T + 31T^{2} \)
37 \( 1 + (-3.5 + 6.06i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (1.5 - 2.59i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (5 + 8.66i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 6T + 47T^{2} \)
53 \( 1 + 3T + 53T^{2} \)
59 \( 1 + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-3.5 - 6.06i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (5 - 8.66i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (3 + 5.19i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + 13T + 73T^{2} \)
79 \( 1 - 4T + 79T^{2} \)
83 \( 1 + 6T + 83T^{2} \)
89 \( 1 + (-9 + 15.5i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (7 + 12.1i)T + (-48.5 + 84.0i)T^{2} \)
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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

−8.874592874455558164594958575140, −8.191755821037741596250100549978, −7.35652493106599782768595313129, −7.10868657182219595592998804249, −5.65217151097424677364641782686, −4.71243225025960076545127876809, −4.36713458918219817682409194842, −2.92216233714403219789188184096, −2.17837658986567269770970508460, −0.23803425182367895221211663084, 1.02212374980078835570853305787, 2.80270353562437943012375696188, 3.54488411526823753165145655514, 4.51031262675736679871533563401, 5.19330429304403538191027186399, 6.34567574085954315672025945844, 7.29885097206527852144155316939, 7.895171759770045984324697481652, 8.376914799379497546863105009751, 9.325773783104568862667596102923

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