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Properties

Label 2-2100-7.6-c2-0-10
Degree $2$
Conductor $2100$
Sign $-0.988 + 0.152i$
Analytic cond. $57.2208$
Root an. cond. $7.56444$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.73i·3-s + (1.06 + 6.91i)7-s − 2.99·9-s + 3.07·11-s + 17.8i·13-s + 24.0i·17-s − 32.7i·19-s + (−11.9 + 1.84i)21-s − 24.3·23-s − 5.19i·27-s + 18.6·29-s + 44.2i·31-s + 5.32i·33-s − 37.2·37-s − 30.9·39-s + ⋯
L(s)  = 1  + 0.577i·3-s + (0.152 + 0.988i)7-s − 0.333·9-s + 0.279·11-s + 1.37i·13-s + 1.41i·17-s − 1.72i·19-s + (−0.570 + 0.0879i)21-s − 1.05·23-s − 0.192i·27-s + 0.643·29-s + 1.42i·31-s + 0.161i·33-s − 1.00·37-s − 0.792·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.988 + 0.152i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.988 + 0.152i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2100\)    =    \(2^{2} \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $-0.988 + 0.152i$
Analytic conductor: \(57.2208\)
Root analytic conductor: \(7.56444\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{2100} (601, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2100,\ (\ :1),\ -0.988 + 0.152i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.129244128\)
\(L(\frac12)\) \(\approx\) \(1.129244128\)
\(L(2)\) 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 \( 1 - 1.73iT \)
5 \( 1 \)
7 \( 1 + (-1.06 - 6.91i)T \)
good11 \( 1 - 3.07T + 121T^{2} \)
13 \( 1 - 17.8iT - 169T^{2} \)
17 \( 1 - 24.0iT - 289T^{2} \)
19 \( 1 + 32.7iT - 361T^{2} \)
23 \( 1 + 24.3T + 529T^{2} \)
29 \( 1 - 18.6T + 841T^{2} \)
31 \( 1 - 44.2iT - 961T^{2} \)
37 \( 1 + 37.2T + 1.36e3T^{2} \)
41 \( 1 - 57.4iT - 1.68e3T^{2} \)
43 \( 1 - 59.9T + 1.84e3T^{2} \)
47 \( 1 + 17.6iT - 2.20e3T^{2} \)
53 \( 1 + 18.8T + 2.80e3T^{2} \)
59 \( 1 + 85.2iT - 3.48e3T^{2} \)
61 \( 1 + 75.7iT - 3.72e3T^{2} \)
67 \( 1 + 29.6T + 4.48e3T^{2} \)
71 \( 1 - 47.5T + 5.04e3T^{2} \)
73 \( 1 + 11.0iT - 5.32e3T^{2} \)
79 \( 1 + 40.4T + 6.24e3T^{2} \)
83 \( 1 - 90.4iT - 6.88e3T^{2} \)
89 \( 1 - 111. iT - 7.92e3T^{2} \)
97 \( 1 + 26.3iT - 9.40e3T^{2} \)
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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

−9.208554567412211452249092741877, −8.724836258810136486554858977896, −8.031273258965980728933150459538, −6.70630833468484672416505561773, −6.31291920491873718761122049366, −5.20328198840464635535233276741, −4.54426022242127811190170067465, −3.64944980908789734164809767437, −2.54306580429100547565853652227, −1.60147846343117064539958544293, 0.28830398537358920175255672639, 1.22695750002964175770601624984, 2.46261136593999908569967515551, 3.53262601802153297927075823767, 4.33606973416952461679875701848, 5.52407348287747754450028001756, 6.07138900919829287763514008886, 7.21730896868576390275732684283, 7.63849391553955448680509613729, 8.273970241513649288231222565049

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