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Properties

Label 2-2100-35.3-c1-0-17
Degree $2$
Conductor $2100$
Sign $0.999 - 0.0337i$
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  + (0.258 + 0.965i)3-s + (1.62 + 2.08i)7-s + (−0.866 + 0.499i)9-s + (0.293 − 0.507i)11-s + (3.67 − 3.67i)13-s + (4.21 − 1.12i)17-s + (−3.38 − 5.86i)19-s + (−1.59 + 2.11i)21-s + (2.30 − 8.61i)23-s + (−0.707 − 0.707i)27-s − 3.32i·29-s + (1.01 + 0.585i)31-s + (0.566 + 0.151i)33-s + (7.06 + 1.89i)37-s + (4.50 + 2.59i)39-s + ⋯
L(s)  = 1  + (0.149 + 0.557i)3-s + (0.615 + 0.788i)7-s + (−0.288 + 0.166i)9-s + (0.0883 − 0.153i)11-s + (1.01 − 1.01i)13-s + (1.02 − 0.273i)17-s + (−0.776 − 1.34i)19-s + (−0.347 + 0.460i)21-s + (0.481 − 1.79i)23-s + (−0.136 − 0.136i)27-s − 0.617i·29-s + (0.182 + 0.105i)31-s + (0.0985 + 0.0264i)33-s + (1.16 + 0.311i)37-s + (0.720 + 0.416i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0337i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0337i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2100\)    =    \(2^{2} \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $0.999 - 0.0337i$
Analytic conductor: \(16.7685\)
Root analytic conductor: \(4.09494\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2100} (493, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2100,\ (\ :1/2),\ 0.999 - 0.0337i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.120276164\)
\(L(\frac12)\) \(\approx\) \(2.120276164\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.258 - 0.965i)T \)
5 \( 1 \)
7 \( 1 + (-1.62 - 2.08i)T \)
good11 \( 1 + (-0.293 + 0.507i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-3.67 + 3.67i)T - 13iT^{2} \)
17 \( 1 + (-4.21 + 1.12i)T + (14.7 - 8.5i)T^{2} \)
19 \( 1 + (3.38 + 5.86i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2.30 + 8.61i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 + 3.32iT - 29T^{2} \)
31 \( 1 + (-1.01 - 0.585i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-7.06 - 1.89i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 - 3.44iT - 41T^{2} \)
43 \( 1 + (-0.0439 - 0.0439i)T + 43iT^{2} \)
47 \( 1 + (-0.195 + 0.731i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (11.3 - 3.05i)T + (45.8 - 26.5i)T^{2} \)
59 \( 1 + (-2.90 + 5.02i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (4.44 - 2.56i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.05 - 11.3i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 - 2.58T + 71T^{2} \)
73 \( 1 + (-2.49 - 9.30i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (-11.8 + 6.83i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-10.6 + 10.6i)T - 83iT^{2} \)
89 \( 1 + (-2.36 - 4.09i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-3.70 - 3.70i)T + 97iT^{2} \)
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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

−9.024712985031539513150434747518, −8.342871751632801609771546362231, −7.914718679103388631493203856547, −6.58502367226048396354063916805, −5.89704248214331151001573334848, −5.01880344308755509414156961073, −4.35844156705153720739699059999, −3.13384852595901663807066961549, −2.46489625750880449842261652039, −0.868576818602482110323271197491, 1.23955023068825254870522351776, 1.83274094736268361160215567220, 3.46733640921228483729473609104, 3.99296548900624193443832699849, 5.14928640024028686984624258395, 6.08146412599298417341085137970, 6.76841519654146739187447465169, 7.79150783634102131575748937549, 7.999556157719894388803703302598, 9.089003188067135968684250278663

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