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Properties

Label 2-2100-35.4-c1-0-6
Degree $2$
Conductor $2100$
Sign $0.330 - 0.943i$
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 + (−0.866 + 2.5i)7-s + (0.499 + 0.866i)9-s + (1 − 1.73i)11-s − i·13-s + (3.46 + 2i)17-s + (−0.5 − 0.866i)19-s + (2 − 1.73i)21-s + (−3.46 + 2i)23-s − 0.999i·27-s + (2.5 − 4.33i)31-s + (−1.73 + 0.999i)33-s + (−4.33 + 2.5i)37-s + (−0.5 + 0.866i)39-s + 2·41-s + ⋯
L(s)  = 1  + (−0.499 − 0.288i)3-s + (−0.327 + 0.944i)7-s + (0.166 + 0.288i)9-s + (0.301 − 0.522i)11-s − 0.277i·13-s + (0.840 + 0.485i)17-s + (−0.114 − 0.198i)19-s + (0.436 − 0.377i)21-s + (−0.722 + 0.417i)23-s − 0.192i·27-s + (0.449 − 0.777i)31-s + (−0.301 + 0.174i)33-s + (−0.711 + 0.410i)37-s + (−0.0800 + 0.138i)39-s + 0.312·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.330 - 0.943i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.330 - 0.943i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2100\)    =    \(2^{2} \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $0.330 - 0.943i$
Analytic conductor: \(16.7685\)
Root analytic conductor: \(4.09494\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2100} (949, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2100,\ (\ :1/2),\ 0.330 - 0.943i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.133418871\)
\(L(\frac12)\) \(\approx\) \(1.133418871\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.866 - 2.5i)T \)
good11 \( 1 + (-1 + 1.73i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + iT - 13T^{2} \)
17 \( 1 + (-3.46 - 2i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.5 + 0.866i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (3.46 - 2i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 + (-2.5 + 4.33i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (4.33 - 2.5i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 - 2T + 41T^{2} \)
43 \( 1 - 9iT - 43T^{2} \)
47 \( 1 + (1.73 - i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-10.3 - 6i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (4 - 6.92i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-7 - 12.1i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (7.79 + 4.5i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 2T + 71T^{2} \)
73 \( 1 + (-0.866 - 0.5i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (1.5 + 2.59i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 18iT - 83T^{2} \)
89 \( 1 + (2 + 3.46i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 10iT - 97T^{2} \)
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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

−9.260750519462560247706680232989, −8.417463303123570320249093456422, −7.78295730085212916845320068170, −6.78665762771505416174433358069, −5.88233401765878038988887266329, −5.65427500396733118386449834260, −4.45065606561756242874149426197, −3.38397863068953463594319108980, −2.40781976714820387002713123017, −1.13490379884284487328547568026, 0.48840492987878236059207745065, 1.83239596464094977389705694104, 3.31027409765475429341831741196, 4.06268993686492117474212392030, 4.87375372662026630917313414699, 5.76332437665598450372719684342, 6.73531594813169951251685606778, 7.18707577113313667564816462881, 8.138121024374544902477008965007, 9.073308800955366273544872869337

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