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Properties

Label 2-2100-21.5-c1-0-10
Degree $2$
Conductor $2100$
Sign $0.643 - 0.765i$
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  + (0.884 − 1.48i)3-s + (−2.64 − 0.0973i)7-s + (−1.43 − 2.63i)9-s + (−2.62 + 1.51i)11-s + 2.31i·13-s + (2.59 + 4.49i)17-s + (5.58 + 3.22i)19-s + (−2.48 + 3.85i)21-s + (−4.21 − 2.43i)23-s + (−5.19 − 0.194i)27-s + 3.48i·29-s + (1.16 − 0.673i)31-s + (−0.0657 + 5.25i)33-s + (−1.40 + 2.43i)37-s + (3.45 + 2.05i)39-s + ⋯
L(s)  = 1  + (0.510 − 0.859i)3-s + (−0.999 − 0.0368i)7-s + (−0.478 − 0.878i)9-s + (−0.792 + 0.457i)11-s + 0.642i·13-s + (0.629 + 1.09i)17-s + (1.28 + 0.739i)19-s + (−0.542 + 0.840i)21-s + (−0.879 − 0.507i)23-s + (−0.999 − 0.0374i)27-s + 0.647i·29-s + (0.209 − 0.120i)31-s + (−0.0114 + 0.915i)33-s + (−0.231 + 0.400i)37-s + (0.552 + 0.328i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.643 - 0.765i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.643 - 0.765i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2100\)    =    \(2^{2} \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $0.643 - 0.765i$
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: \((2,\ 2100,\ (\ :1/2),\ 0.643 - 0.765i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.264707883\)
\(L(\frac12)\) \(\approx\) \(1.264707883\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.884 + 1.48i)T \)
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

−9.194996409594250928027453127179, −8.290091383490949356862963328600, −7.67390364034920974494162537661, −6.94916036524814041457345242014, −6.16534388010961502458887564743, −5.48772156804449071245524440414, −4.08063011507072667816895219168, −3.27737907967213177046369333856, −2.37185134366737834180306253647, −1.24473312792127115062808479108, 0.43441392058922970509881191735, 2.49632889627694398320090083652, 3.15116821930403051236542721696, 3.84235514521381831664326556434, 5.23056981088100383818325247285, 5.43846024750813330259925821999, 6.67698414619022596793047828357, 7.68687970254146371166587657967, 8.164270644456685353983034166220, 9.269148785444099910687195087966

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