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Properties

Label 2-2100-7.2-c1-0-12
Degree $2$
Conductor $2100$
Sign $0.832 - 0.553i$
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 + (1.32 − 2.29i)7-s + (−0.499 + 0.866i)9-s + (0.822 + 1.42i)11-s + 2.64·13-s + (−0.822 − 1.42i)17-s + (−4.14 + 7.18i)19-s + 2.64·21-s + (0.822 − 1.42i)23-s − 0.999·27-s + 7.64·29-s + (2.14 + 3.71i)31-s + (−0.822 + 1.42i)33-s + (0.322 − 0.559i)37-s + (1.32 + 2.29i)39-s + ⋯
L(s)  = 1  + (0.288 + 0.499i)3-s + (0.499 − 0.866i)7-s + (−0.166 + 0.288i)9-s + (0.248 + 0.429i)11-s + 0.733·13-s + (−0.199 − 0.345i)17-s + (−0.951 + 1.64i)19-s + 0.577·21-s + (0.171 − 0.297i)23-s − 0.192·27-s + 1.41·29-s + (0.385 + 0.667i)31-s + (−0.143 + 0.248i)33-s + (0.0530 − 0.0919i)37-s + (0.211 + 0.366i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.832 - 0.553i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.832 - 0.553i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2100\)    =    \(2^{2} \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $0.832 - 0.553i$
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: \((2,\ 2100,\ (\ :1/2),\ 0.832 - 0.553i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.155173615\)
\(L(\frac12)\) \(\approx\) \(2.155173615\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.32 + 2.29i)T \)
good11 \( 1 + (-0.822 - 1.42i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 2.64T + 13T^{2} \)
17 \( 1 + (0.822 + 1.42i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (4.14 - 7.18i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.822 + 1.42i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 7.64T + 29T^{2} \)
31 \( 1 + (-2.14 - 3.71i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.322 + 0.559i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 4.93T + 41T^{2} \)
43 \( 1 - 5.93T + 43T^{2} \)
47 \( 1 + (-3 + 5.19i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-1.64 - 2.85i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (5.46 + 9.47i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (4 - 6.92i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.322 - 0.559i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 13.6T + 71T^{2} \)
73 \( 1 + (-6.61 - 11.4i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (1.14 - 1.98i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 10.9T + 83T^{2} \)
89 \( 1 + (-7.11 + 12.3i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 8T + 97T^{2} \)
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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

−9.126370349127816045845212839023, −8.364481055327640828176746709575, −7.81535376542627738936291086173, −6.82569659254001288274509164766, −6.08714186704864156519377879029, −4.95797595604611718842242042495, −4.21503280202259462139539935015, −3.60032505420382364134945238221, −2.30814256297129907361802393111, −1.11398651682934146680990877536, 0.910013020525929562130353992357, 2.18889918748144899683826263925, 2.92577324543165517348446567507, 4.15769982052311540891757990430, 5.01383023649767751770725840721, 6.13548369069432546898939869506, 6.48222300275815638737762001224, 7.62129421119247916229837675473, 8.358917002212911497023538770608, 8.900370752210743745685198717892

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