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Properties

Label 2-2100-35.3-c1-0-23
Degree $2$
Conductor $2100$
Sign $-0.957 + 0.287i$
Analytic cond. $16.7685$
Root an. cond. $4.09494$
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  + (−0.258 − 0.965i)3-s + (−0.920 − 2.48i)7-s + (−0.866 + 0.499i)9-s + (1.65 − 2.87i)11-s + (1.05 − 1.05i)13-s + (−3.20 + 0.858i)17-s + (1.10 + 1.91i)19-s + (−2.15 + 1.53i)21-s + (1.83 − 6.83i)23-s + (0.707 + 0.707i)27-s + (1.56 + 0.902i)31-s + (−3.20 − 0.858i)33-s + (−1.67 − 0.448i)37-s + (−1.29 − 0.747i)39-s − 1.33i·41-s + ⋯
L(s)  = 1  + (−0.149 − 0.557i)3-s + (−0.348 − 0.937i)7-s + (−0.288 + 0.166i)9-s + (0.499 − 0.865i)11-s + (0.293 − 0.293i)13-s + (−0.776 + 0.208i)17-s + (0.253 + 0.438i)19-s + (−0.470 + 0.334i)21-s + (0.381 − 1.42i)23-s + (0.136 + 0.136i)27-s + (0.280 + 0.162i)31-s + (−0.557 − 0.149i)33-s + (−0.275 − 0.0736i)37-s + (−0.207 − 0.119i)39-s − 0.207i·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.957 + 0.287i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.957 + 0.287i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2100\)    =    \(2^{2} \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $-0.957 + 0.287i$
Analytic conductor: \(16.7685\)
Root analytic conductor: \(4.09494\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2100} (493, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2100,\ (\ :1/2),\ -0.957 + 0.287i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.014258616\)
\(L(\frac12)\) \(\approx\) \(1.014258616\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.258 + 0.965i)T \)
5 \( 1 \)
7 \( 1 + (0.920 + 2.48i)T \)
good11 \( 1 + (-1.65 + 2.87i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-1.05 + 1.05i)T - 13iT^{2} \)
17 \( 1 + (3.20 - 0.858i)T + (14.7 - 8.5i)T^{2} \)
19 \( 1 + (-1.10 - 1.91i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.83 + 6.83i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 - 29T^{2} \)
31 \( 1 + (-1.56 - 0.902i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (1.67 + 0.448i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + 1.33iT - 41T^{2} \)
43 \( 1 + (6.84 + 6.84i)T + 43iT^{2} \)
47 \( 1 + (1.11 - 4.17i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (2.97 - 0.797i)T + (45.8 - 26.5i)T^{2} \)
59 \( 1 + (0.665 - 1.15i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (4.62 - 2.67i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.570 - 2.13i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 - 4.98T + 71T^{2} \)
73 \( 1 + (2.74 + 10.2i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (13.8 - 7.96i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-10.1 + 10.1i)T - 83iT^{2} \)
89 \( 1 + (-2.87 - 4.97i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-3.53 - 3.53i)T + 97iT^{2} \)
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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

−8.606174373076225762621461617683, −8.039510711119173145141399387558, −7.01370148815051496465714594198, −6.52819743408351431884167783620, −5.77618282169078667177777619191, −4.64372475395606930533564016999, −3.74654327767125245678132333817, −2.85323047646330341113338640013, −1.46734154897901476928327021511, −0.36945970835139492626916699678, 1.63862037043108780518401995748, 2.77646428002068359103620935396, 3.72338121241829898890228433949, 4.71252106697683119914145290369, 5.36353856276633787523219717289, 6.35870597956734732318840329333, 6.93962062467689399733437827749, 8.016493047994011747738632652392, 8.940867858960928680576662602275, 9.430176710296957693410284723091

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