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Properties

Label 2-2100-35.33-c1-0-13
Degree $2$
Conductor $2100$
Sign $0.762 + 0.647i$
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  + (−0.965 − 0.258i)3-s + (2.02 − 1.70i)7-s + (0.866 + 0.499i)9-s + (1.21 + 2.10i)11-s + (−0.728 + 0.728i)13-s + (1.96 − 7.33i)17-s + (−1.84 + 3.20i)19-s + (−2.39 + 1.12i)21-s + (2.83 − 0.759i)23-s + (−0.707 − 0.707i)27-s + 1.99i·29-s + (6.02 − 3.47i)31-s + (−0.629 − 2.34i)33-s + (2.20 + 8.22i)37-s + (0.892 − 0.515i)39-s + ⋯
L(s)  = 1  + (−0.557 − 0.149i)3-s + (0.763 − 0.645i)7-s + (0.288 + 0.166i)9-s + (0.366 + 0.634i)11-s + (−0.202 + 0.202i)13-s + (0.476 − 1.77i)17-s + (−0.424 + 0.734i)19-s + (−0.522 + 0.245i)21-s + (0.591 − 0.158i)23-s + (−0.136 − 0.136i)27-s + 0.369i·29-s + (1.08 − 0.624i)31-s + (−0.109 − 0.408i)33-s + (0.362 + 1.35i)37-s + (0.142 − 0.0825i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.762 + 0.647i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.762 + 0.647i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2100\)    =    \(2^{2} \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $0.762 + 0.647i$
Analytic conductor: \(16.7685\)
Root analytic conductor: \(4.09494\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2100} (1993, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2100,\ (\ :1/2),\ 0.762 + 0.647i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.610758772\)
\(L(\frac12)\) \(\approx\) \(1.610758772\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.965 + 0.258i)T \)
5 \( 1 \)
7 \( 1 + (-2.02 + 1.70i)T \)
good11 \( 1 + (-1.21 - 2.10i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (0.728 - 0.728i)T - 13iT^{2} \)
17 \( 1 + (-1.96 + 7.33i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (1.84 - 3.20i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.83 + 0.759i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 - 1.99iT - 29T^{2} \)
31 \( 1 + (-6.02 + 3.47i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2.20 - 8.22i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 - 1.08iT - 41T^{2} \)
43 \( 1 + (5.91 + 5.91i)T + 43iT^{2} \)
47 \( 1 + (-4.81 + 1.28i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (1.76 - 6.56i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (-1.39 - 2.41i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.217 - 0.125i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (7.79 + 2.08i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 - 14.5T + 71T^{2} \)
73 \( 1 + (6.46 + 1.73i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (9.54 + 5.51i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-9.90 + 9.90i)T - 83iT^{2} \)
89 \( 1 + (-5.81 + 10.0i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-10.8 - 10.8i)T + 97iT^{2} \)
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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

−9.060990403176443104628864899284, −8.093739549766789189042592937318, −7.33304379597644836763373046556, −6.84389860967304396905965029327, −5.83509356263810238757212784748, −4.79232972425356060793898558800, −4.48206011574806116956180258052, −3.15296301509208903082920467869, −1.87532368219638205267281291086, −0.77976680134755986903084668747, 1.04719613501661367139941216772, 2.23261875232250444449360408298, 3.46557316539297207961067299604, 4.43327959766263065676773025365, 5.25934067512025086858915090004, 5.99475078705457885139226785823, 6.64166461914529672392991682950, 7.78753475180106014048710075965, 8.433611837090826154886858021434, 9.081710926294012724521208776729

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