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Properties

Label 2-1872-13.3-c1-0-0
Degree $2$
Conductor $1872$
Sign $-0.993 - 0.116i$
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  − 0.561·5-s + (−1.78 + 3.08i)7-s + (1 + 1.73i)11-s + (0.5 + 3.57i)13-s + (1.28 − 2.21i)17-s + (−0.561 + 0.972i)19-s + (−1 − 1.73i)23-s − 4.68·25-s + (−2.84 − 4.92i)29-s + 1.56·31-s + (1 − 1.73i)35-s + (−1.71 − 2.97i)37-s + (1.28 + 2.21i)41-s + (0.219 − 0.379i)43-s − 8.24·47-s + ⋯
L(s)  = 1  − 0.251·5-s + (−0.673 + 1.16i)7-s + (0.301 + 0.522i)11-s + (0.138 + 0.990i)13-s + (0.310 − 0.538i)17-s + (−0.128 + 0.223i)19-s + (−0.208 − 0.361i)23-s − 0.936·25-s + (−0.527 − 0.914i)29-s + 0.280·31-s + (0.169 − 0.292i)35-s + (−0.282 − 0.489i)37-s + (0.200 + 0.346i)41-s + (0.0334 − 0.0579i)43-s − 1.20·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.993 - 0.116i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.993 - 0.116i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1872\)    =    \(2^{4} \cdot 3^{2} \cdot 13\)
Sign: $-0.993 - 0.116i$
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: \((2,\ 1872,\ (\ :1/2),\ -0.993 - 0.116i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5236929475\)
\(L(\frac12)\) \(\approx\) \(0.5236929475\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 - 3.57i)T \)
good5 \( 1 + 0.561T + 5T^{2} \)
7 \( 1 + (1.78 - 3.08i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-1 - 1.73i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (-1.28 + 2.21i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.561 - 0.972i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (1 + 1.73i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (2.84 + 4.92i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 1.56T + 31T^{2} \)
37 \( 1 + (1.71 + 2.97i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-1.28 - 2.21i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.219 + 0.379i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 8.24T + 47T^{2} \)
53 \( 1 + 11.6T + 53T^{2} \)
59 \( 1 + (-5.56 + 9.63i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (6.06 - 10.4i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.219 - 0.379i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (7 - 12.1i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + 1.87T + 73T^{2} \)
79 \( 1 + 9.56T + 79T^{2} \)
83 \( 1 + 9.12T + 83T^{2} \)
89 \( 1 + (-6.56 - 11.3i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-2.21 + 3.84i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.569697305045175133359875654815, −8.936760692936271062393130415204, −8.104322452112368002540746453983, −7.21061767524825018746577110178, −6.35180616339952177303933917014, −5.75783397113832675577623537599, −4.66599582314206090072485485401, −3.81351453256690141878979734106, −2.72315625193968901039322420666, −1.78143906619449626558213197266, 0.18967071343113966316784322037, 1.47361621579349193060777081583, 3.19166903804453655247433322401, 3.62931988204294652891298826562, 4.65065332309894890780288677607, 5.77125295850033826516540376345, 6.44881644393682657849242837723, 7.38547630859108491840798157651, 7.945737180173613100178995825843, 8.823793401809679759004479734590

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