Properties

Label 2-1872-13.3-c1-0-22
Degree $2$
Conductor $1872$
Sign $0.522 + 0.852i$
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  + 5-s + (−1 + 1.73i)7-s + (−1 − 1.73i)11-s + (2.5 − 2.59i)13-s + (2.5 − 4.33i)17-s + (−1 + 1.73i)19-s + (−3 − 5.19i)23-s − 4·25-s + (−4.5 − 7.79i)29-s + 4·31-s + (−1 + 1.73i)35-s + (5.5 + 9.52i)37-s + (2.5 + 4.33i)41-s + (5 − 8.66i)43-s + 2·47-s + ⋯
L(s)  = 1  + 0.447·5-s + (−0.377 + 0.654i)7-s + (−0.301 − 0.522i)11-s + (0.693 − 0.720i)13-s + (0.606 − 1.05i)17-s + (−0.229 + 0.397i)19-s + (−0.625 − 1.08i)23-s − 0.800·25-s + (−0.835 − 1.44i)29-s + 0.718·31-s + (−0.169 + 0.292i)35-s + (0.904 + 1.56i)37-s + (0.390 + 0.676i)41-s + (0.762 − 1.32i)43-s + 0.291·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.522 + 0.852i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.522 + 0.852i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1872\)    =    \(2^{4} \cdot 3^{2} \cdot 13\)
Sign: $0.522 + 0.852i$
Analytic conductor: \(14.9479\)
Root analytic conductor: \(3.86626\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1872} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1872,\ (\ :1/2),\ 0.522 + 0.852i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.605937833\)
\(L(\frac12)\) \(\approx\) \(1.605937833\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.5 + 2.59i)T \)
good5 \( 1 - T + 5T^{2} \)
7 \( 1 + (1 - 1.73i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (1 + 1.73i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (-2.5 + 4.33i)T + (-8.5 - 14.7i)T^{2} \)
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 \( 1 + (4.5 + 7.79i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 4T + 31T^{2} \)
37 \( 1 + (-5.5 - 9.52i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-2.5 - 4.33i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-5 + 8.66i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 2T + 47T^{2} \)
53 \( 1 - T + 53T^{2} \)
59 \( 1 + (-4 + 6.92i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-5.5 + 9.52i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1 - 1.73i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-7 + 12.1i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + 13T + 73T^{2} \)
79 \( 1 - 4T + 79T^{2} \)
83 \( 1 - 6T + 83T^{2} \)
89 \( 1 + (-1 - 1.73i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-1 + 1.73i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.201060259375020521338433812087, −8.188592751639197024399381725221, −7.81401513061245207078217938453, −6.42169582931045086417112206243, −5.97113420228711341236747556333, −5.24985249861799050676876762351, −4.07564093734086821180278664756, −3.01894452984742968709996788637, −2.22968338850981852345911785895, −0.62970035992411405556549463418, 1.27678710269308195306696795851, 2.31765351092679198261158164338, 3.70527061135221080198398406082, 4.19402699793562034886300302776, 5.53881216575934154401572640544, 6.06578816714884024643003101581, 7.08944851368932495615367363725, 7.64487036034322077934396723802, 8.664640280902247966873422335453, 9.441776299637401590771252595340

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