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Properties

Label 2-1860-31.25-c1-0-5
Degree $2$
Conductor $1860$
Sign $-0.640 - 0.767i$
Analytic cond. $14.8521$
Root an. cond. $3.85385$
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  + (0.5 + 0.866i)3-s + (−0.5 + 0.866i)5-s + (1.87 + 3.24i)7-s + (−0.499 + 0.866i)9-s + (−2.24 + 3.89i)11-s + (2.50 − 4.33i)13-s − 0.999·15-s + (−1.17 − 2.02i)17-s + (3.50 + 6.07i)19-s + (−1.87 + 3.24i)21-s + 6.06·23-s + (−0.499 − 0.866i)25-s − 0.999·27-s − 8.35·29-s + (1.61 + 5.32i)31-s + ⋯
L(s)  = 1  + (0.288 + 0.499i)3-s + (−0.223 + 0.387i)5-s + (0.708 + 1.22i)7-s + (−0.166 + 0.288i)9-s + (−0.677 + 1.17i)11-s + (0.694 − 1.20i)13-s − 0.258·15-s + (−0.284 − 0.492i)17-s + (0.804 + 1.39i)19-s + (−0.408 + 0.708i)21-s + 1.26·23-s + (−0.0999 − 0.173i)25-s − 0.192·27-s − 1.55·29-s + (0.290 + 0.956i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1860 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.640 - 0.767i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1860 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.640 - 0.767i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1860\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 31\)
Sign: $-0.640 - 0.767i$
Analytic conductor: \(14.8521\)
Root analytic conductor: \(3.85385\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1860} (1141, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1860,\ (\ :1/2),\ -0.640 - 0.767i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.749928235\)
\(L(\frac12)\) \(\approx\) \(1.749928235\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 - 0.866i)T \)
5 \( 1 + (0.5 - 0.866i)T \)
31 \( 1 + (-1.61 - 5.32i)T \)
good7 \( 1 + (-1.87 - 3.24i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (2.24 - 3.89i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-2.50 + 4.33i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (1.17 + 2.02i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3.50 - 6.07i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 6.06T + 23T^{2} \)
29 \( 1 + 8.35T + 29T^{2} \)
37 \( 1 + (-0.873 - 1.51i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-1.94 + 3.37i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.931 - 1.61i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 3.40T + 47T^{2} \)
53 \( 1 + (2.90 - 5.03i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (4.05 + 7.02i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + 13.3T + 61T^{2} \)
67 \( 1 + (1.34 - 2.33i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (2.96 - 5.14i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-8.20 + 14.2i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (4.16 + 7.21i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (2.16 - 3.74i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 13.5T + 89T^{2} \)
97 \( 1 - 4.31T + 97T^{2} \)
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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

−9.416021665487119635657487174918, −8.820108531767840691659302398853, −7.80601006759255550022263508633, −7.57027276249759898737378311406, −6.16835479898998344244660638063, −5.30209491912078637966448466705, −4.84053574063122196040193332512, −3.50127211064877830272822943334, −2.77363253219300097283075040193, −1.68933925057077363732023412066, 0.63673578120553434146136664896, 1.61041966553173728845021799974, 2.99894734516263879892210494603, 3.97713408972867645139903902802, 4.74930275350126113196396250274, 5.76645365762442911298135567983, 6.77339303159649095576605325331, 7.45935454312569515987436335840, 8.091480931682487848482024259010, 8.904706769299123467472168107333

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