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Properties

Label 2-1872-13.4-c1-0-21
Degree $2$
Conductor $1872$
Sign $-0.227 + 0.973i$
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  + (−4.5 + 2.59i)7-s + (−2.5 + 2.59i)13-s + (3 − 1.73i)19-s + 5·25-s − 8.66i·31-s + (−6 − 3.46i)37-s + (−6.5 − 11.2i)43-s + (10 − 17.3i)49-s + (−6.5 − 11.2i)61-s + (10.5 + 6.06i)67-s + 1.73i·73-s − 13·79-s + (4.5 − 18.1i)91-s + (−16.5 + 9.52i)97-s + 13·103-s + ⋯
L(s)  = 1  + (−1.70 + 0.981i)7-s + (−0.693 + 0.720i)13-s + (0.688 − 0.397i)19-s + 25-s − 1.55i·31-s + (−0.986 − 0.569i)37-s + (−0.991 − 1.71i)43-s + (1.42 − 2.47i)49-s + (−0.832 − 1.44i)61-s + (1.28 + 0.740i)67-s + 0.202i·73-s − 1.46·79-s + (0.471 − 1.90i)91-s + (−1.67 + 0.967i)97-s + 1.28·103-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.227 + 0.973i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.227 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1872\)    =    \(2^{4} \cdot 3^{2} \cdot 13\)
Sign: $-0.227 + 0.973i$
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: \((2,\ 1872,\ (\ :1/2),\ -0.227 + 0.973i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5254547477\)
\(L(\frac12)\) \(\approx\) \(0.5254547477\)
\(L(\frac{3}{2})\) 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 \( 1 - 5T^{2} \)
7 \( 1 + (4.5 - 2.59i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3 + 1.73i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 8.66iT - 31T^{2} \)
37 \( 1 + (6 + 3.46i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (6.5 + 11.2i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 47T^{2} \)
53 \( 1 + 53T^{2} \)
59 \( 1 + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (6.5 + 11.2i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-10.5 - 6.06i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (35.5 - 61.4i)T^{2} \)
73 \( 1 - 1.73iT - 73T^{2} \)
79 \( 1 + 13T + 79T^{2} \)
83 \( 1 - 83T^{2} \)
89 \( 1 + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (16.5 - 9.52i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.191828061747497082956040237728, −8.415828913659778977367154055652, −7.14951754536012708732776646912, −6.73439692079206936648473774906, −5.80869676095609006725140305897, −5.09010904166559153909563342951, −3.85673828228338419254397576347, −2.97947230813279821631773916377, −2.14947045124176899687830411569, −0.20903242480536027883324706855, 1.14214071696002958645883226899, 2.98776449439593926433129434646, 3.32246481759121778041600236476, 4.52000787618939290002677192193, 5.44932253342109961711376108924, 6.48911821246867901283856990383, 6.99117279212361253848842526326, 7.74533690991497304844805343422, 8.761757896854679554068394627585, 9.625404835215112473600895791159

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