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Properties

Label 2-1860-155.92-c1-0-8
Degree $2$
Conductor $1860$
Sign $0.449 - 0.893i$
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 + (−2.03 + 0.930i)5-s + (3.55 + 3.55i)7-s + 1.00i·9-s − 1.02i·11-s + (−2.14 − 2.14i)13-s + (2.09 + 0.779i)15-s + (4.10 − 4.10i)17-s + 1.55i·19-s − 5.02i·21-s + (1.29 + 1.29i)23-s + (3.26 − 3.78i)25-s + (0.707 − 0.707i)27-s + 5.13·29-s + (−3.42 + 4.38i)31-s + ⋯
L(s)  = 1  + (−0.408 − 0.408i)3-s + (−0.909 + 0.416i)5-s + (1.34 + 1.34i)7-s + 0.333i·9-s − 0.308i·11-s + (−0.594 − 0.594i)13-s + (0.541 + 0.201i)15-s + (0.996 − 0.996i)17-s + 0.356i·19-s − 1.09i·21-s + (0.269 + 0.269i)23-s + (0.653 − 0.756i)25-s + (0.136 − 0.136i)27-s + 0.953·29-s + (−0.615 + 0.787i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1860 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.449 - 0.893i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1860 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.449 - 0.893i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1860\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 31\)
Sign: $0.449 - 0.893i$
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: \((2,\ 1860,\ (\ :1/2),\ 0.449 - 0.893i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.307702672\)
\(L(\frac12)\) \(\approx\) \(1.307702672\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.03 - 0.930i)T \)
31 \( 1 + (3.42 - 4.38i)T \)
good7 \( 1 + (-3.55 - 3.55i)T + 7iT^{2} \)
11 \( 1 + 1.02iT - 11T^{2} \)
13 \( 1 + (2.14 + 2.14i)T + 13iT^{2} \)
17 \( 1 + (-4.10 + 4.10i)T - 17iT^{2} \)
19 \( 1 - 1.55iT - 19T^{2} \)
23 \( 1 + (-1.29 - 1.29i)T + 23iT^{2} \)
29 \( 1 - 5.13T + 29T^{2} \)
37 \( 1 + (8.19 - 8.19i)T - 37iT^{2} \)
41 \( 1 - 3.51T + 41T^{2} \)
43 \( 1 + (-3.18 - 3.18i)T + 43iT^{2} \)
47 \( 1 + (-4.33 - 4.33i)T + 47iT^{2} \)
53 \( 1 + (6.42 + 6.42i)T + 53iT^{2} \)
59 \( 1 + 1.91iT - 59T^{2} \)
61 \( 1 - 4.45iT - 61T^{2} \)
67 \( 1 + (-0.870 - 0.870i)T + 67iT^{2} \)
71 \( 1 + 9.24T + 71T^{2} \)
73 \( 1 + (-2.35 - 2.35i)T + 73iT^{2} \)
79 \( 1 - 3.74T + 79T^{2} \)
83 \( 1 + (-4.05 - 4.05i)T + 83iT^{2} \)
89 \( 1 - 13.1T + 89T^{2} \)
97 \( 1 + (-5.44 - 5.44i)T + 97iT^{2} \)
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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

−9.212741242357090357087871776450, −8.339370526807365936375724126646, −7.85969737509912137701320165731, −7.19129013807018905836717106624, −6.12574968844082823686305515466, −5.17951344446777282985871673390, −4.83394455964947735715263073427, −3.30374019323674074300033937084, −2.49939218882073799526431769198, −1.15372653824062541559867355456, 0.60706308991288880920144253205, 1.77355217281482450651144608211, 3.55755765418719986038232646827, 4.30854805568149670528604686024, 4.73937471914715949982073689736, 5.67719016070062609719696946502, 7.06172914560582125629486921723, 7.46523211725120177756091271871, 8.227586821413436901728475148740, 9.021746939346034250390509841369

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