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Properties

Label 2-2100-105.89-c1-0-42
Degree $2$
Conductor $2100$
Sign $-0.162 + 0.986i$
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  + (1.42 − 0.979i)3-s + (−2.60 − 0.456i)7-s + (1.08 − 2.79i)9-s + (0.698 − 0.403i)11-s + 3.86·13-s + (1.83 − 1.05i)17-s + (−2.70 − 1.56i)19-s + (−4.17 + 1.89i)21-s + (2.43 − 4.21i)23-s + (−1.19 − 5.05i)27-s + 6.67i·29-s + (5.65 − 3.26i)31-s + (0.603 − 1.25i)33-s + (−6.75 − 3.89i)37-s + (5.52 − 3.78i)39-s + ⋯
L(s)  = 1  + (0.824 − 0.565i)3-s + (−0.985 − 0.172i)7-s + (0.360 − 0.932i)9-s + (0.210 − 0.121i)11-s + 1.07·13-s + (0.445 − 0.257i)17-s + (−0.620 − 0.358i)19-s + (−0.910 + 0.414i)21-s + (0.508 − 0.879i)23-s + (−0.229 − 0.973i)27-s + 1.23i·29-s + (1.01 − 0.585i)31-s + (0.104 − 0.219i)33-s + (−1.11 − 0.641i)37-s + (0.884 − 0.606i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.162 + 0.986i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.162 + 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2100\)    =    \(2^{2} \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $-0.162 + 0.986i$
Analytic conductor: \(16.7685\)
Root analytic conductor: \(4.09494\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2100} (1349, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2100,\ (\ :1/2),\ -0.162 + 0.986i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.019181304\)
\(L(\frac12)\) \(\approx\) \(2.019181304\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.42 + 0.979i)T \)
5 \( 1 \)
7 \( 1 + (2.60 + 0.456i)T \)
good11 \( 1 + (-0.698 + 0.403i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 - 3.86T + 13T^{2} \)
17 \( 1 + (-1.83 + 1.05i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.70 + 1.56i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2.43 + 4.21i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 6.67iT - 29T^{2} \)
31 \( 1 + (-5.65 + 3.26i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (6.75 + 3.89i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + 8.44T + 41T^{2} \)
43 \( 1 - 0.819iT - 43T^{2} \)
47 \( 1 + (-2.40 - 1.38i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (6.67 + 11.5i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (2.86 + 4.95i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.79 - 1.03i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-9.44 + 5.45i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 10.3iT - 71T^{2} \)
73 \( 1 + (1.54 + 2.67i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (6.76 - 11.7i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 12.8iT - 83T^{2} \)
89 \( 1 + (-1.60 + 2.78i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 1.01T + 97T^{2} \)
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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

−8.725805633907913377571361188407, −8.343699355821929045156873772427, −7.21065508979270138771586943249, −6.64450710618897194130842248498, −6.04659116311687031612663799693, −4.74188509750963228668858381892, −3.57530868479686881170693113686, −3.15037844393724604821854598223, −1.92972840979252677378065665515, −0.65745886728036815670319648611, 1.47330525879940067220485613975, 2.75704080138400849979496003916, 3.52659516118635625900067206467, 4.17731102198136371905941927252, 5.32201164412169535825124200477, 6.20485425743668292658974644698, 6.97437987757773105734603314164, 8.000033667462519528692028190951, 8.629564446596003786668730428730, 9.253893735346570472100288358947

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