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Properties

Label 2-1860-155.92-c1-0-2
Degree $2$
Conductor $1860$
Sign $-0.502 - 0.864i$
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  + (−0.707 − 0.707i)3-s + (1.97 + 1.04i)5-s + (−0.798 − 0.798i)7-s + 1.00i·9-s + 5.73i·11-s + (−1.95 − 1.95i)13-s + (−0.663 − 2.13i)15-s + (−3.32 + 3.32i)17-s − 6.00i·19-s + 1.12i·21-s + (5.81 + 5.81i)23-s + (2.83 + 4.11i)25-s + (0.707 − 0.707i)27-s − 9.28·29-s + (−5.26 − 1.79i)31-s + ⋯
L(s)  = 1  + (−0.408 − 0.408i)3-s + (0.885 + 0.465i)5-s + (−0.301 − 0.301i)7-s + 0.333i·9-s + 1.73i·11-s + (−0.541 − 0.541i)13-s + (−0.171 − 0.551i)15-s + (−0.806 + 0.806i)17-s − 1.37i·19-s + 0.246i·21-s + (1.21 + 1.21i)23-s + (0.566 + 0.823i)25-s + (0.136 − 0.136i)27-s − 1.72·29-s + (−0.946 − 0.322i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1860 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.502 - 0.864i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1860 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.502 - 0.864i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1860\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 31\)
Sign: $-0.502 - 0.864i$
Analytic conductor: \(14.8521\)
Root analytic conductor: \(3.85385\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1860} (1177, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1860,\ (\ :1/2),\ -0.502 - 0.864i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8465996884\)
\(L(\frac12)\) \(\approx\) \(0.8465996884\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.707 + 0.707i)T \)
5 \( 1 + (-1.97 - 1.04i)T \)
31 \( 1 + (5.26 + 1.79i)T \)
good7 \( 1 + (0.798 + 0.798i)T + 7iT^{2} \)
11 \( 1 - 5.73iT - 11T^{2} \)
13 \( 1 + (1.95 + 1.95i)T + 13iT^{2} \)
17 \( 1 + (3.32 - 3.32i)T - 17iT^{2} \)
19 \( 1 + 6.00iT - 19T^{2} \)
23 \( 1 + (-5.81 - 5.81i)T + 23iT^{2} \)
29 \( 1 + 9.28T + 29T^{2} \)
37 \( 1 + (6.10 - 6.10i)T - 37iT^{2} \)
41 \( 1 + 9.94T + 41T^{2} \)
43 \( 1 + (0.0243 + 0.0243i)T + 43iT^{2} \)
47 \( 1 + (-8.37 - 8.37i)T + 47iT^{2} \)
53 \( 1 + (3.37 + 3.37i)T + 53iT^{2} \)
59 \( 1 - 0.400iT - 59T^{2} \)
61 \( 1 + 8.24iT - 61T^{2} \)
67 \( 1 + (-5.55 - 5.55i)T + 67iT^{2} \)
71 \( 1 + 6.44T + 71T^{2} \)
73 \( 1 + (-6.75 - 6.75i)T + 73iT^{2} \)
79 \( 1 - 0.429T + 79T^{2} \)
83 \( 1 + (1.28 + 1.28i)T + 83iT^{2} \)
89 \( 1 + 3.49T + 89T^{2} \)
97 \( 1 + (0.568 + 0.568i)T + 97iT^{2} \)
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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

−9.561985092742756775408359746832, −8.949070713533356656415559901421, −7.54123982664994907078916275098, −7.06537477080418575060685640063, −6.55407934480726851411444595976, −5.37531871949345005849640019053, −4.90318343355177094311354630432, −3.58346631228874802169089083703, −2.37521324752741389871917748990, −1.61246047281030273960887649958, 0.30534787490760176900294760574, 1.83373482791098225009734262093, 3.00917321453439059546175201588, 4.00928309047892041042050530648, 5.18249232095752804513402128878, 5.61499659456138975602022469195, 6.41153367178775517422429762331, 7.23617297266675052369714295735, 8.627751424182518902619540531955, 8.948262990926522267838639769199

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