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Properties

Label 2-1872-13.4-c1-0-6
Degree $2$
Conductor $1872$
Sign $0.265 - 0.964i$
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  − 3.73i·5-s + (−2.36 + 1.36i)7-s + (1.09 + 0.633i)11-s + (−2.59 + 2.5i)13-s + (2.86 + 4.96i)17-s + (−4.09 + 2.36i)19-s + (2.09 − 3.63i)23-s − 8.92·25-s + (−2.23 + 3.86i)29-s − 1.46i·31-s + (5.09 + 8.83i)35-s + (3.06 + 1.76i)37-s + (−8.13 − 4.69i)41-s + (4.83 + 8.36i)43-s − 2.19i·47-s + ⋯
L(s)  = 1  − 1.66i·5-s + (−0.894 + 0.516i)7-s + (0.331 + 0.191i)11-s + (−0.720 + 0.693i)13-s + (0.695 + 1.20i)17-s + (−0.940 + 0.542i)19-s + (0.437 − 0.757i)23-s − 1.78·25-s + (−0.414 + 0.717i)29-s − 0.262i·31-s + (0.861 + 1.49i)35-s + (0.503 + 0.290i)37-s + (−1.27 − 0.733i)41-s + (0.736 + 1.27i)43-s − 0.320i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.265 - 0.964i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.265 - 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1872\)    =    \(2^{4} \cdot 3^{2} \cdot 13\)
Sign: $0.265 - 0.964i$
Analytic conductor: \(14.9479\)
Root analytic conductor: \(3.86626\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1872} (433, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1872,\ (\ :1/2),\ 0.265 - 0.964i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8900275008\)
\(L(\frac12)\) \(\approx\) \(0.8900275008\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.59 - 2.5i)T \)
good5 \( 1 + 3.73iT - 5T^{2} \)
7 \( 1 + (2.36 - 1.36i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-1.09 - 0.633i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (-2.86 - 4.96i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (4.09 - 2.36i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.09 + 3.63i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (2.23 - 3.86i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 1.46iT - 31T^{2} \)
37 \( 1 + (-3.06 - 1.76i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (8.13 + 4.69i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-4.83 - 8.36i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 2.19iT - 47T^{2} \)
53 \( 1 - 6.46T + 53T^{2} \)
59 \( 1 + (6.92 - 4i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4.59 - 7.96i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-11.3 - 6.56i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-4.09 + 2.36i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 - 6.26iT - 73T^{2} \)
79 \( 1 - 2.53T + 79T^{2} \)
83 \( 1 + 0.196iT - 83T^{2} \)
89 \( 1 + (-8.19 - 4.73i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (5.19 - 3i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.346186156061080273315979023549, −8.643283884392320091544962346986, −8.126711545095316445409112120493, −6.94384481378956860495678577596, −6.10319589174468896248413298343, −5.36713891312074962798839302427, −4.45967987150735224544544887405, −3.75712707449025414375243915440, −2.31863657989459619960299699898, −1.23330870726592085771285865537, 0.34156125650097746099352801886, 2.34667998785262292360818744669, 3.14276534913239257187332781737, 3.73781223880333018593921508335, 5.07717583428523280393702250212, 6.08173485506847640583162557992, 6.86332458161479582813863519860, 7.23549015905815311025147182605, 8.065915433573776297068582115981, 9.421111605227269641797062503631

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