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Properties

Label 2-2100-7.2-c1-0-18
Degree $2$
Conductor $2100$
Sign $-0.266 + 0.963i$
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  + (−0.5 − 0.866i)3-s + (−0.5 − 2.59i)7-s + (−0.499 + 0.866i)9-s + (−1 − 1.73i)11-s + 3·13-s + (4 + 6.92i)17-s + (0.5 − 0.866i)19-s + (−2 + 1.73i)21-s + (4 − 6.92i)23-s + 0.999·27-s + 4·29-s + (−1.5 − 2.59i)31-s + (−0.999 + 1.73i)33-s + (−0.5 + 0.866i)37-s + (−1.5 − 2.59i)39-s + ⋯
L(s)  = 1  + (−0.288 − 0.499i)3-s + (−0.188 − 0.981i)7-s + (−0.166 + 0.288i)9-s + (−0.301 − 0.522i)11-s + 0.832·13-s + (0.970 + 1.68i)17-s + (0.114 − 0.198i)19-s + (−0.436 + 0.377i)21-s + (0.834 − 1.44i)23-s + 0.192·27-s + 0.742·29-s + (−0.269 − 0.466i)31-s + (−0.174 + 0.301i)33-s + (−0.0821 + 0.142i)37-s + (−0.240 − 0.416i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.266 + 0.963i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.266 + 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2100\)    =    \(2^{2} \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $-0.266 + 0.963i$
Analytic conductor: \(16.7685\)
Root analytic conductor: \(4.09494\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2100} (1801, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2100,\ (\ :1/2),\ -0.266 + 0.963i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.425872785\)
\(L(\frac12)\) \(\approx\) \(1.425872785\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.5 + 0.866i)T \)
5 \( 1 \)
7 \( 1 + (0.5 + 2.59i)T \)
good11 \( 1 + (1 + 1.73i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 3T + 13T^{2} \)
17 \( 1 + (-4 - 6.92i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.5 + 0.866i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-4 + 6.92i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 4T + 29T^{2} \)
31 \( 1 + (1.5 + 2.59i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.5 - 0.866i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 6T + 41T^{2} \)
43 \( 1 + 11T + 43T^{2} \)
47 \( 1 + (-3 + 5.19i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (6 + 10.3i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (2 + 3.46i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-3 + 5.19i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-6.5 - 11.2i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 10T + 71T^{2} \)
73 \( 1 + (5.5 + 9.52i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-1.5 + 2.59i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 2T + 83T^{2} \)
89 \( 1 + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 10T + 97T^{2} \)
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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

−8.481586197286060665889451892708, −8.274031303280547287898977168784, −7.24994931249509686195318535508, −6.48508810262852846364315702894, −5.91976027743212472565729749444, −4.86976500263577129667346499165, −3.86458612992151212977773671285, −3.07471891287172468221521771786, −1.61882765419548335065548026140, −0.59146379977460636388211260175, 1.26499302594765782524353377093, 2.76403516598285403654098936602, 3.39885405141085957937069191497, 4.68342841456055938379128876223, 5.35265945924216109852996909503, 5.96596966402527894882673227731, 7.01362081111772859095173615222, 7.75881040428835068151252853549, 8.756491253430226388308794240206, 9.404867300453660491687727090464

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