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Properties

Label 2-2100-35.9-c1-0-23
Degree $2$
Conductor $2100$
Sign $-0.990 + 0.138i$
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 + (−0.358 − 2.62i)7-s + (0.499 − 0.866i)9-s + (−2.12 − 3.67i)11-s − 5i·13-s + (3.67 − 2.12i)17-s + (−2.62 + 4.54i)19-s + (1.62 + 2.09i)21-s + (5.19 + 3i)23-s + 0.999i·27-s − 8.48·29-s + (2 + 3.46i)31-s + (3.67 + 2.12i)33-s + (−4.33 − 2.5i)37-s + (2.5 + 4.33i)39-s + ⋯
L(s)  = 1  + (−0.499 + 0.288i)3-s + (−0.135 − 0.990i)7-s + (0.166 − 0.288i)9-s + (−0.639 − 1.10i)11-s − 1.38i·13-s + (0.891 − 0.514i)17-s + (−0.601 + 1.04i)19-s + (0.353 + 0.456i)21-s + (1.08 + 0.625i)23-s + 0.192i·27-s − 1.57·29-s + (0.359 + 0.622i)31-s + (0.639 + 0.369i)33-s + (−0.711 − 0.410i)37-s + (0.400 + 0.693i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.990 + 0.138i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.990 + 0.138i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2100\)    =    \(2^{2} \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $-0.990 + 0.138i$
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: \((2,\ 2100,\ (\ :1/2),\ -0.990 + 0.138i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4479114310\)
\(L(\frac12)\) \(\approx\) \(0.4479114310\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.358 + 2.62i)T \)
good11 \( 1 + (2.12 + 3.67i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 5iT - 13T^{2} \)
17 \( 1 + (-3.67 + 2.12i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.62 - 4.54i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-5.19 - 3i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 8.48T + 29T^{2} \)
31 \( 1 + (-2 - 3.46i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (4.33 + 2.5i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + 10.2T + 41T^{2} \)
43 \( 1 - 12.4iT - 43T^{2} \)
47 \( 1 + (8.87 + 5.12i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (3.67 - 2.12i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-5.12 - 8.87i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.5 + 0.866i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-10.1 + 5.86i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 8.48T + 71T^{2} \)
73 \( 1 + (-3.01 + 1.74i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-4.62 + 8.00i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 1.75iT - 83T^{2} \)
89 \( 1 + (-3.87 + 6.71i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 17.9iT - 97T^{2} \)
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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

−8.631928397067684524377051440510, −7.88849144886522659966462080034, −7.28541706838733120553429015955, −6.22948468157316392912329096476, −5.48647293571803889434272766591, −4.89056261797786401990098242417, −3.47834307595929072538854844109, −3.24450951430932292387481420343, −1.32469830716755580428127488127, −0.17223945806252837757544083537, 1.74847173226090150153339233080, 2.44922320104061297909254333497, 3.78973044629388717339428297976, 4.95105098373043561353283434030, 5.31762027335480360183082281967, 6.58776419000087437845733678606, 6.85706612202977519764811801990, 7.914501288484074272433780867691, 8.750913864345769612454158321149, 9.452669653082220385100731183971

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