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Properties

Label 2-2100-21.5-c1-0-1
Degree $2$
Conductor $2100$
Sign $-0.518 - 0.855i$
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  + (0.317 − 1.70i)3-s + (1.73 − 1.99i)7-s + (−2.79 − 1.07i)9-s + (−3.38 + 1.95i)11-s + 6.06i·13-s + (−1.53 − 2.65i)17-s + (−2.94 − 1.70i)19-s + (−2.85 − 3.58i)21-s + (−2.48 − 1.43i)23-s + (−2.72 + 4.42i)27-s + 7.97i·29-s + (−5.63 + 3.25i)31-s + (2.25 + 6.37i)33-s + (0.0654 − 0.113i)37-s + (10.3 + 1.92i)39-s + ⋯
L(s)  = 1  + (0.183 − 0.983i)3-s + (0.655 − 0.755i)7-s + (−0.932 − 0.359i)9-s + (−1.01 + 0.588i)11-s + 1.68i·13-s + (−0.371 − 0.643i)17-s + (−0.676 − 0.390i)19-s + (−0.622 − 0.782i)21-s + (−0.518 − 0.299i)23-s + (−0.524 + 0.851i)27-s + 1.48i·29-s + (−1.01 + 0.583i)31-s + (0.391 + 1.10i)33-s + (0.0107 − 0.0186i)37-s + (1.65 + 0.307i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.518 - 0.855i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.518 - 0.855i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2100\)    =    \(2^{2} \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $-0.518 - 0.855i$
Analytic conductor: \(16.7685\)
Root analytic conductor: \(4.09494\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2100} (1601, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2100,\ (\ :1/2),\ -0.518 - 0.855i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.08657550238\)
\(L(\frac12)\) \(\approx\) \(0.08657550238\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.317 + 1.70i)T \)
5 \( 1 \)
7 \( 1 + (-1.73 + 1.99i)T \)
good11 \( 1 + (3.38 - 1.95i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 - 6.06iT - 13T^{2} \)
17 \( 1 + (1.53 + 2.65i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.94 + 1.70i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (2.48 + 1.43i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 7.97iT - 29T^{2} \)
31 \( 1 + (5.63 - 3.25i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-0.0654 + 0.113i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 12.3T + 41T^{2} \)
43 \( 1 + 4.43T + 43T^{2} \)
47 \( 1 + (-5.02 + 8.71i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (4.64 - 2.67i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-1.28 - 2.23i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-7.44 - 4.30i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (7.99 + 13.8i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 3.63iT - 71T^{2} \)
73 \( 1 + (-6.72 + 3.88i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (1.22 - 2.11i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 7.63T + 83T^{2} \)
89 \( 1 + (-4.11 + 7.13i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 5.74iT - 97T^{2} \)
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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

−9.071893272375651039173540261272, −8.586176953636724186164513047441, −7.67428957875443230607950374326, −6.95608238380813662327631327889, −6.68346874207124987270320523135, −5.26799755437532991264272921159, −4.64398742397466965054881826651, −3.53370618996914649101189234498, −2.23241590930605065093517846975, −1.62111025326252298387080878818, 0.02691464020884598805858231456, 2.09915556593792866302037843236, 2.96876460266206239962583114660, 3.87702341207260856848963161117, 4.90354741639171894506709255619, 5.62484196552386734122552000107, 6.03862150012964529450697624376, 7.70469444154693905955065416619, 8.266139000912499095911029497210, 8.602099903765147240420450609936

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