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Properties

Label 2-2100-21.5-c1-0-27
Degree $2$
Conductor $2100$
Sign $0.988 + 0.150i$
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  + (−1.73 + 0.0216i)3-s + (2.64 + 0.0973i)7-s + (2.99 − 0.0749i)9-s + (2.62 − 1.51i)11-s − 2.31i·13-s + (2.59 + 4.49i)17-s + (5.58 + 3.22i)19-s + (−4.58 − 0.111i)21-s + (−4.21 − 2.43i)23-s + (−5.19 + 0.194i)27-s − 3.48i·29-s + (1.16 − 0.673i)31-s + (−4.52 + 2.68i)33-s + (1.40 − 2.43i)37-s + (0.0501 + 4.01i)39-s + ⋯
L(s)  = 1  + (−0.999 + 0.0124i)3-s + (0.999 + 0.0368i)7-s + (0.999 − 0.0249i)9-s + (0.792 − 0.457i)11-s − 0.642i·13-s + (0.629 + 1.09i)17-s + (1.28 + 0.739i)19-s + (−0.999 − 0.0243i)21-s + (−0.879 − 0.507i)23-s + (−0.999 + 0.0374i)27-s − 0.647i·29-s + (0.209 − 0.120i)31-s + (−0.786 + 0.467i)33-s + (0.231 − 0.400i)37-s + (0.00803 + 0.642i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.988 + 0.150i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.988 + 0.150i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2100\)    =    \(2^{2} \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $0.988 + 0.150i$
Analytic conductor: \(16.7685\)
Root analytic conductor: \(4.09494\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2100} (1601, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2100,\ (\ :1/2),\ 0.988 + 0.150i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.587668129\)
\(L(\frac12)\) \(\approx\) \(1.587668129\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.73 - 0.0216i)T \)
5 \( 1 \)
7 \( 1 + (-2.64 - 0.0973i)T \)
good11 \( 1 + (-2.62 + 1.51i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + 2.31iT - 13T^{2} \)
17 \( 1 + (-2.59 - 4.49i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-5.58 - 3.22i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (4.21 + 2.43i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 3.48iT - 29T^{2} \)
31 \( 1 + (-1.16 + 0.673i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1.40 + 2.43i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 1.06T + 41T^{2} \)
43 \( 1 + 2.42T + 43T^{2} \)
47 \( 1 + (1.90 - 3.30i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (6.11 - 3.52i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (6.18 + 10.7i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-11.4 - 6.60i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.14 + 3.72i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 7.08iT - 71T^{2} \)
73 \( 1 + (0.132 - 0.0763i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-3.04 + 5.27i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 10.2T + 83T^{2} \)
89 \( 1 + (-7.10 + 12.3i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 8.63iT - 97T^{2} \)
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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

−9.153749835318282961272404695392, −7.949246215970590889850237460212, −7.81469155178523592959054831241, −6.50048973058921374516075806482, −5.88129018160128197689449557093, −5.23511149221286141968029889349, −4.27255360822106937244416474778, −3.47923203732569809063258158053, −1.82537548755106696259447529768, −0.909360930457597723955542637645, 0.979436566753245479013592819685, 1.89918643788382804246493809808, 3.44494483103533946552477893518, 4.56403671531440046137479025140, 5.02617572762926501898004969794, 5.86430547985532501506076437140, 6.93639864091092866080028913107, 7.30449001232220780393227128340, 8.251597125214503740261499496671, 9.410878172248298855817394282429

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