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Properties

Label 2-2100-105.59-c1-0-7
Degree $2$
Conductor $2100$
Sign $-0.470 - 0.882i$
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  + (−1.64 + 0.546i)3-s + (2.41 + 1.08i)7-s + (2.40 − 1.79i)9-s + (−1.17 − 0.675i)11-s − 4.94·13-s + (4.97 + 2.87i)17-s + (−2.84 + 1.64i)19-s + (−4.55 − 0.460i)21-s + (2.50 + 4.33i)23-s + (−2.96 + 4.26i)27-s − 5.68i·29-s + (−2.45 − 1.41i)31-s + (2.29 + 0.471i)33-s + (−3.33 + 1.92i)37-s + (8.12 − 2.69i)39-s + ⋯
L(s)  = 1  + (−0.948 + 0.315i)3-s + (0.912 + 0.409i)7-s + (0.801 − 0.598i)9-s + (−0.353 − 0.203i)11-s − 1.37·13-s + (1.20 + 0.696i)17-s + (−0.653 + 0.377i)19-s + (−0.994 − 0.100i)21-s + (0.521 + 0.903i)23-s + (−0.571 + 0.820i)27-s − 1.05i·29-s + (−0.440 − 0.254i)31-s + (0.399 + 0.0820i)33-s + (−0.548 + 0.316i)37-s + (1.30 − 0.432i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.470 - 0.882i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.470 - 0.882i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2100\)    =    \(2^{2} \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $-0.470 - 0.882i$
Analytic conductor: \(16.7685\)
Root analytic conductor: \(4.09494\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2100} (1949, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2100,\ (\ :1/2),\ -0.470 - 0.882i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8935467985\)
\(L(\frac12)\) \(\approx\) \(0.8935467985\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.64 - 0.546i)T \)
5 \( 1 \)
7 \( 1 + (-2.41 - 1.08i)T \)
good11 \( 1 + (1.17 + 0.675i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + 4.94T + 13T^{2} \)
17 \( 1 + (-4.97 - 2.87i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.84 - 1.64i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.50 - 4.33i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 5.68iT - 29T^{2} \)
31 \( 1 + (2.45 + 1.41i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (3.33 - 1.92i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 - 3.73T + 41T^{2} \)
43 \( 1 - 4.06iT - 43T^{2} \)
47 \( 1 + (-4.92 + 2.84i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-0.730 + 1.26i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (4.34 - 7.52i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.65 + 0.954i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-4.36 - 2.51i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 3.38iT - 71T^{2} \)
73 \( 1 + (8.16 - 14.1i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-2.41 - 4.18i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 16.6iT - 83T^{2} \)
89 \( 1 + (8.08 + 13.9i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 12.0T + 97T^{2} \)
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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

−9.624080848637578218641292031733, −8.520294403607588192101776605950, −7.73107874629690028797841909918, −7.09474909545809272670352095575, −5.86690121857109744666606242895, −5.48326549944369276924699862140, −4.66931213753414921602443329530, −3.81454663581099432565284068446, −2.46383221530904026233740840943, −1.25941679661512371591132922015, 0.38547566013158534955126691181, 1.65342652610385882856336754047, 2.75268993232918636068085515224, 4.24195201429831291072936594618, 5.03361694037685577111269576380, 5.38437281382004542325925709566, 6.63841934012747060666062599180, 7.34889289825172528389883353632, 7.73285023503289737224785510312, 8.831766065175755197051713358702

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