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Properties

Label 2-1860-155.123-c1-0-29
Degree $2$
Conductor $1860$
Sign $-0.0509 + 0.998i$
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 + (1.82 − 1.29i)5-s + (2.05 − 2.05i)7-s − 1.00i·9-s − 1.00i·11-s + (1.61 − 1.61i)13-s + (0.374 − 2.20i)15-s + (−3.86 − 3.86i)17-s + 5.41i·19-s − 2.91i·21-s + (0.0230 − 0.0230i)23-s + (1.64 − 4.72i)25-s + (−0.707 − 0.707i)27-s + 1.52·29-s + (4.36 − 3.46i)31-s + ⋯
L(s)  = 1  + (0.408 − 0.408i)3-s + (0.815 − 0.578i)5-s + (0.777 − 0.777i)7-s − 0.333i·9-s − 0.303i·11-s + (0.449 − 0.449i)13-s + (0.0966 − 0.569i)15-s + (−0.937 − 0.937i)17-s + 1.24i·19-s − 0.635i·21-s + (0.00480 − 0.00480i)23-s + (0.329 − 0.944i)25-s + (−0.136 − 0.136i)27-s + 0.282·29-s + (0.783 − 0.621i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1860 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0509 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1860 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0509 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1860\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 31\)
Sign: $-0.0509 + 0.998i$
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: \((2,\ 1860,\ (\ :1/2),\ -0.0509 + 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.496930897\)
\(L(\frac12)\) \(\approx\) \(2.496930897\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.82 + 1.29i)T \)
31 \( 1 + (-4.36 + 3.46i)T \)
good7 \( 1 + (-2.05 + 2.05i)T - 7iT^{2} \)
11 \( 1 + 1.00iT - 11T^{2} \)
13 \( 1 + (-1.61 + 1.61i)T - 13iT^{2} \)
17 \( 1 + (3.86 + 3.86i)T + 17iT^{2} \)
19 \( 1 - 5.41iT - 19T^{2} \)
23 \( 1 + (-0.0230 + 0.0230i)T - 23iT^{2} \)
29 \( 1 - 1.52T + 29T^{2} \)
37 \( 1 + (3.04 + 3.04i)T + 37iT^{2} \)
41 \( 1 - 4.90T + 41T^{2} \)
43 \( 1 + (3.37 - 3.37i)T - 43iT^{2} \)
47 \( 1 + (6.30 - 6.30i)T - 47iT^{2} \)
53 \( 1 + (8.56 - 8.56i)T - 53iT^{2} \)
59 \( 1 + 8.14iT - 59T^{2} \)
61 \( 1 - 14.6iT - 61T^{2} \)
67 \( 1 + (-5.16 + 5.16i)T - 67iT^{2} \)
71 \( 1 - 8.00T + 71T^{2} \)
73 \( 1 + (4.34 - 4.34i)T - 73iT^{2} \)
79 \( 1 - 3.07T + 79T^{2} \)
83 \( 1 + (-10.8 + 10.8i)T - 83iT^{2} \)
89 \( 1 - 2.30T + 89T^{2} \)
97 \( 1 + (-1.43 + 1.43i)T - 97iT^{2} \)
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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

−9.005949685041568998421467173534, −8.141499633184536806164108657257, −7.71605358937084792412968951017, −6.59606752813134229410434730160, −5.91264430505244548136309714085, −4.90627316886892379351780179301, −4.15202103294177032424854182626, −2.91881333470194232884586052217, −1.80366007976492786348912490869, −0.893180251965237798840728507554, 1.75550120598944938668595460946, 2.40412452800771063553904914506, 3.48140527769119510871753268521, 4.67162398993017308245961977231, 5.25338530519991009541937101880, 6.40202732269867247023877844522, 6.85331287934517625925574381680, 8.167361831696330650860631172970, 8.681315214304465292804647340724, 9.380350496915439019602162297757

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