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Properties

Label 2-1860-155.123-c1-0-11
Degree $2$
Conductor $1860$
Sign $0.857 - 0.514i$
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 + (1.61 − 1.54i)5-s + (−2.80 + 2.80i)7-s − 1.00i·9-s − 2.34i·11-s + (−0.590 + 0.590i)13-s + (−0.0455 + 2.23i)15-s + (3.30 + 3.30i)17-s − 6.38i·19-s − 3.97i·21-s + (−3.47 + 3.47i)23-s + (0.203 − 4.99i)25-s + (0.707 + 0.707i)27-s + 8.33·29-s + (4.99 + 2.45i)31-s + ⋯
L(s)  = 1  + (−0.408 + 0.408i)3-s + (0.721 − 0.692i)5-s + (−1.06 + 1.06i)7-s − 0.333i·9-s − 0.706i·11-s + (−0.163 + 0.163i)13-s + (−0.0117 + 0.577i)15-s + (0.801 + 0.801i)17-s − 1.46i·19-s − 0.866i·21-s + (−0.724 + 0.724i)23-s + (0.0407 − 0.999i)25-s + (0.136 + 0.136i)27-s + 1.54·29-s + (0.897 + 0.440i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1860 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.857 - 0.514i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1860 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.857 - 0.514i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1860\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 31\)
Sign: $0.857 - 0.514i$
Analytic conductor: \(14.8521\)
Root analytic conductor: \(3.85385\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1860} (433, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1860,\ (\ :1/2),\ 0.857 - 0.514i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.463069566\)
\(L(\frac12)\) \(\approx\) \(1.463069566\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.61 + 1.54i)T \)
31 \( 1 + (-4.99 - 2.45i)T \)
good7 \( 1 + (2.80 - 2.80i)T - 7iT^{2} \)
11 \( 1 + 2.34iT - 11T^{2} \)
13 \( 1 + (0.590 - 0.590i)T - 13iT^{2} \)
17 \( 1 + (-3.30 - 3.30i)T + 17iT^{2} \)
19 \( 1 + 6.38iT - 19T^{2} \)
23 \( 1 + (3.47 - 3.47i)T - 23iT^{2} \)
29 \( 1 - 8.33T + 29T^{2} \)
37 \( 1 + (-3.67 - 3.67i)T + 37iT^{2} \)
41 \( 1 - 4.78T + 41T^{2} \)
43 \( 1 + (7.19 - 7.19i)T - 43iT^{2} \)
47 \( 1 + (-7.44 + 7.44i)T - 47iT^{2} \)
53 \( 1 + (7.23 - 7.23i)T - 53iT^{2} \)
59 \( 1 - 11.9iT - 59T^{2} \)
61 \( 1 - 4.05iT - 61T^{2} \)
67 \( 1 + (-7.19 + 7.19i)T - 67iT^{2} \)
71 \( 1 - 2.52T + 71T^{2} \)
73 \( 1 + (-10.7 + 10.7i)T - 73iT^{2} \)
79 \( 1 + 3.70T + 79T^{2} \)
83 \( 1 + (-2.11 + 2.11i)T - 83iT^{2} \)
89 \( 1 - 18.5T + 89T^{2} \)
97 \( 1 + (-1.23 + 1.23i)T - 97iT^{2} \)
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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

−9.272139491523759361367530010631, −8.804472783936854072437603492092, −7.934710849531287190075914187909, −6.39372783307580885355988508674, −6.21683225659212607365348119593, −5.32649967581128650244133831260, −4.58502775111252088095443725182, −3.30440875123099643831552075716, −2.47933641428494137799858147354, −0.931450436855664661059411569523, 0.75798733211150590757332593116, 2.16016365907150253686254577246, 3.16883196315765647291677396645, 4.14220983420434409487679209749, 5.27453342493438747016446883424, 6.26363474382190592537699201743, 6.64085145110043073561563452340, 7.45692277392677287125027362333, 8.127018527224416659813080390648, 9.629525189162417095737585726616

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