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Properties

Label 2-2100-105.104-c1-0-38
Degree $2$
Conductor $2100$
Sign $-0.998 + 0.0460i$
Analytic cond. $16.7685$
Root an. cond. $4.09494$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.61 + 0.618i)3-s + (−0.381 − 2.61i)7-s + (2.23 − 2.00i)9-s + 5.23i·11-s − 3.23·13-s − 4.47i·17-s − 2.76i·19-s + (2.23 + 4i)21-s + 5.70·23-s + (−2.38 + 4.61i)27-s + 4i·29-s + 1.23i·31-s + (−3.23 − 8.47i)33-s − 4.47i·37-s + (5.23 − 2.00i)39-s + ⋯
L(s)  = 1  + (−0.934 + 0.356i)3-s + (−0.144 − 0.989i)7-s + (0.745 − 0.666i)9-s + 1.57i·11-s − 0.897·13-s − 1.08i·17-s − 0.634i·19-s + (0.487 + 0.872i)21-s + 1.19·23-s + (−0.458 + 0.888i)27-s + 0.742i·29-s + 0.222i·31-s + (−0.563 − 1.47i)33-s − 0.735i·37-s + (0.838 − 0.320i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 + 0.0460i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.998 + 0.0460i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2100\)    =    \(2^{2} \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $-0.998 + 0.0460i$
Analytic conductor: \(16.7685\)
Root analytic conductor: \(4.09494\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2100} (1049, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 2100,\ (\ :1/2),\ -0.998 + 0.0460i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.61 - 0.618i)T \)
5 \( 1 \)
7 \( 1 + (0.381 + 2.61i)T \)
good11 \( 1 - 5.23iT - 11T^{2} \)
13 \( 1 + 3.23T + 13T^{2} \)
17 \( 1 + 4.47iT - 17T^{2} \)
19 \( 1 + 2.76iT - 19T^{2} \)
23 \( 1 - 5.70T + 23T^{2} \)
29 \( 1 - 4iT - 29T^{2} \)
31 \( 1 - 1.23iT - 31T^{2} \)
37 \( 1 + 4.47iT - 37T^{2} \)
41 \( 1 + 12.4T + 41T^{2} \)
43 \( 1 - 2.76iT - 43T^{2} \)
47 \( 1 - 1.23iT - 47T^{2} \)
53 \( 1 + 4.76T + 53T^{2} \)
59 \( 1 - 8.94T + 59T^{2} \)
61 \( 1 - 12.9iT - 61T^{2} \)
67 \( 1 + 3.70iT - 67T^{2} \)
71 \( 1 + 10.1iT - 71T^{2} \)
73 \( 1 + 11.2T + 73T^{2} \)
79 \( 1 + 1.52T + 79T^{2} \)
83 \( 1 - 2.76iT - 83T^{2} \)
89 \( 1 - 3.52T + 89T^{2} \)
97 \( 1 + 8.18T + 97T^{2} \)
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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

−9.034707915157893419415248129676, −7.53006471553669153924157857054, −7.07243902726877980122671513361, −6.64795859632820579460090235973, −5.10540898097233351690398310490, −4.91318966044700894105267236931, −4.02151417175022910978778983921, −2.82006796447401445450936316266, −1.36432430034131910361612659743, 0, 1.47128435472330521694695715814, 2.66496122735462218499956333935, 3.72509765451001374746363636441, 4.99952184686913169152352280075, 5.57767936108639944451595296675, 6.25655267201982712301319864296, 6.92423347412695661690225523395, 8.113396197181382838652532643591, 8.487362323209727631900064041753

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