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Properties

Label 2-1860-155.123-c1-0-20
Degree $2$
Conductor $1860$
Sign $0.680 + 0.733i$
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 + (−1.64 + 1.51i)5-s + (0.674 − 0.674i)7-s − 1.00i·9-s + 1.09i·11-s + (2.63 − 2.63i)13-s + (0.0885 − 2.23i)15-s + (−1.41 − 1.41i)17-s − 2.52i·19-s + 0.954i·21-s + (−5.53 + 5.53i)23-s + (0.395 − 4.98i)25-s + (0.707 + 0.707i)27-s − 5.18·29-s + (1.26 − 5.42i)31-s + ⋯
L(s)  = 1  + (−0.408 + 0.408i)3-s + (−0.734 + 0.678i)5-s + (0.255 − 0.255i)7-s − 0.333i·9-s + 0.330i·11-s + (0.730 − 0.730i)13-s + (0.0228 − 0.576i)15-s + (−0.343 − 0.343i)17-s − 0.580i·19-s + 0.208i·21-s + (−1.15 + 1.15i)23-s + (0.0791 − 0.996i)25-s + (0.136 + 0.136i)27-s − 0.963·29-s + (0.227 − 0.973i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1860 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.680 + 0.733i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1860 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.680 + 0.733i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1860\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 31\)
Sign: $0.680 + 0.733i$
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: \((2,\ 1860,\ (\ :1/2),\ 0.680 + 0.733i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9483283474\)
\(L(\frac12)\) \(\approx\) \(0.9483283474\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.64 - 1.51i)T \)
31 \( 1 + (-1.26 + 5.42i)T \)
good7 \( 1 + (-0.674 + 0.674i)T - 7iT^{2} \)
11 \( 1 - 1.09iT - 11T^{2} \)
13 \( 1 + (-2.63 + 2.63i)T - 13iT^{2} \)
17 \( 1 + (1.41 + 1.41i)T + 17iT^{2} \)
19 \( 1 + 2.52iT - 19T^{2} \)
23 \( 1 + (5.53 - 5.53i)T - 23iT^{2} \)
29 \( 1 + 5.18T + 29T^{2} \)
37 \( 1 + (3.34 + 3.34i)T + 37iT^{2} \)
41 \( 1 - 4.37T + 41T^{2} \)
43 \( 1 + (3.16 - 3.16i)T - 43iT^{2} \)
47 \( 1 + (-7.37 + 7.37i)T - 47iT^{2} \)
53 \( 1 + (-3.45 + 3.45i)T - 53iT^{2} \)
59 \( 1 + 11.6iT - 59T^{2} \)
61 \( 1 + 4.88iT - 61T^{2} \)
67 \( 1 + (-3.63 + 3.63i)T - 67iT^{2} \)
71 \( 1 - 15.2T + 71T^{2} \)
73 \( 1 + (-5.18 + 5.18i)T - 73iT^{2} \)
79 \( 1 - 8.53T + 79T^{2} \)
83 \( 1 + (0.755 - 0.755i)T - 83iT^{2} \)
89 \( 1 + 6.58T + 89T^{2} \)
97 \( 1 + (-4.81 + 4.81i)T - 97iT^{2} \)
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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

−9.285684763244443575472916913021, −8.155623449407357637728394279719, −7.62316534927324697704568120484, −6.78327275555115754505893298361, −5.92344544099428460531689357262, −5.05535629832304058597855254993, −4.00245601430057178470943953791, −3.49108708222124211023135875921, −2.16738677585721365951586203280, −0.43479628178288554703635805571, 1.07345197928827129378694097547, 2.17542977977043809449448490169, 3.70559446386007420739566357228, 4.35425933381673387115408534025, 5.36465712108332925034082977384, 6.13231212401564984436485305852, 6.95453803565098662996232141233, 7.899556016514265451265947465431, 8.526635971838730117439161436849, 9.036133036904013462638847468650

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