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Properties

Label 2-2100-35.9-c1-0-3
Degree $2$
Conductor $2100$
Sign $-0.502 - 0.864i$
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  + (0.866 − 0.5i)3-s + (−2.29 + 1.32i)7-s + (0.499 − 0.866i)9-s + (2.82 + 4.88i)11-s − i·13-s + (−1.42 + 0.822i)17-s + (−2.32 + 4.02i)19-s + (−1.32 + 2.29i)21-s + (−1.73 − i)23-s − 0.999i·27-s − 7.29·29-s + (−2 − 3.46i)31-s + (4.88 + 2.82i)33-s + (−7.18 − 4.14i)37-s + (−0.5 − 0.866i)39-s + ⋯
L(s)  = 1  + (0.499 − 0.288i)3-s + (−0.866 + 0.499i)7-s + (0.166 − 0.288i)9-s + (0.851 + 1.47i)11-s − 0.277i·13-s + (−0.345 + 0.199i)17-s + (−0.532 + 0.923i)19-s + (−0.288 + 0.499i)21-s + (−0.361 − 0.208i)23-s − 0.192i·27-s − 1.35·29-s + (−0.359 − 0.622i)31-s + (0.851 + 0.491i)33-s + (−1.18 − 0.681i)37-s + (−0.0800 − 0.138i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.502 - 0.864i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.502 - 0.864i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2100\)    =    \(2^{2} \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $-0.502 - 0.864i$
Analytic conductor: \(16.7685\)
Root analytic conductor: \(4.09494\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2100} (1549, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2100,\ (\ :1/2),\ -0.502 - 0.864i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.051431750\)
\(L(\frac12)\) \(\approx\) \(1.051431750\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.866 + 0.5i)T \)
5 \( 1 \)
7 \( 1 + (2.29 - 1.32i)T \)
good11 \( 1 + (-2.82 - 4.88i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + iT - 13T^{2} \)
17 \( 1 + (1.42 - 0.822i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.32 - 4.02i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.73 + i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 7.29T + 29T^{2} \)
31 \( 1 + (2 + 3.46i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (7.18 + 4.14i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + 0.354T + 41T^{2} \)
43 \( 1 - 7.29iT - 43T^{2} \)
47 \( 1 + (2.54 + 1.46i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (2.03 - 1.17i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-7.46 - 12.9i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (4.79 - 8.29i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (6.87 - 3.96i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 3.29T + 71T^{2} \)
73 \( 1 + (-5.44 + 3.14i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (3.32 - 5.75i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 3.64iT - 83T^{2} \)
89 \( 1 + (4.46 - 7.73i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 14.2iT - 97T^{2} \)
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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

−9.330723665939869027605832193227, −8.724689485732085499767387704224, −7.74211428904548813235684540355, −7.05351834331199476844624521697, −6.32711596027997774538673230280, −5.54283244837212331042783841575, −4.25256889734253236701836573097, −3.65563447493681376645943168181, −2.43570937737053422381897483694, −1.65740504793405531130324965387, 0.32511517072534225414977889917, 1.87736694880716792811330296751, 3.27178015669380520981350131728, 3.61396798787077943010467746831, 4.66243666216259456313374335811, 5.75647373053051635302474139772, 6.60391663189128365569658595936, 7.15459900205721740233741836478, 8.282975296760102563893656884910, 8.940486521271109747358680505447

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