Properties

Label 2-2100-21.20-c1-0-33
Degree $2$
Conductor $2100$
Sign $-0.0515 + 0.998i$
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 − 1.41i)3-s + (−2.23 + 1.41i)7-s + (−1.00 − 2.82i)9-s + 2.82i·11-s + 2.82i·13-s + 4·17-s − 6.32i·19-s + (−0.236 + 4.57i)21-s − 6.32i·23-s + (−5.00 − 1.41i)27-s − 5.65i·29-s − 6.32i·31-s + (4.00 + 2.82i)33-s + 8.94·37-s + (4.00 + 2.82i)39-s + ⋯
L(s)  = 1  + (0.577 − 0.816i)3-s + (−0.845 + 0.534i)7-s + (−0.333 − 0.942i)9-s + 0.852i·11-s + 0.784i·13-s + 0.970·17-s − 1.45i·19-s + (−0.0515 + 0.998i)21-s − 1.31i·23-s + (−0.962 − 0.272i)27-s − 1.05i·29-s − 1.13i·31-s + (0.696 + 0.492i)33-s + 1.47·37-s + (0.640 + 0.452i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0515 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0515 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2100\)    =    \(2^{2} \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $-0.0515 + 0.998i$
Analytic conductor: \(16.7685\)
Root analytic conductor: \(4.09494\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2100} (1301, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2100,\ (\ :1/2),\ -0.0515 + 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.704201034\)
\(L(\frac12)\) \(\approx\) \(1.704201034\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1 + 1.41i)T \)
5 \( 1 \)
7 \( 1 + (2.23 - 1.41i)T \)
good11 \( 1 - 2.82iT - 11T^{2} \)
13 \( 1 - 2.82iT - 13T^{2} \)
17 \( 1 - 4T + 17T^{2} \)
19 \( 1 + 6.32iT - 19T^{2} \)
23 \( 1 + 6.32iT - 23T^{2} \)
29 \( 1 + 5.65iT - 29T^{2} \)
31 \( 1 + 6.32iT - 31T^{2} \)
37 \( 1 - 8.94T + 37T^{2} \)
41 \( 1 + 4.47T + 41T^{2} \)
43 \( 1 - 4.47T + 43T^{2} \)
47 \( 1 - 6T + 47T^{2} \)
53 \( 1 + 6.32iT - 53T^{2} \)
59 \( 1 + 8.94T + 59T^{2} \)
61 \( 1 - 61T^{2} \)
67 \( 1 - 4.47T + 67T^{2} \)
71 \( 1 + 14.1iT - 71T^{2} \)
73 \( 1 + 8.48iT - 73T^{2} \)
79 \( 1 - 12T + 79T^{2} \)
83 \( 1 + 10T + 83T^{2} \)
89 \( 1 + 4.47T + 89T^{2} \)
97 \( 1 - 8.48iT - 97T^{2} \)
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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

−9.093726516102119278882378030508, −8.023556842419293705497796831902, −7.38602873218251697817370182729, −6.52010210179974891981294856239, −6.10761152260331602636290958478, −4.81588145847436760520036011753, −3.85421372371903882723711967465, −2.69728818112346763175590312435, −2.17486628800189304740489406178, −0.59906208980625665939897346550, 1.25301348025842130937741264451, 2.98750901658983860713578081899, 3.38632192914247224779042283096, 4.18664215981407121322338641075, 5.50160908928694922955219877676, 5.83505520196398056942323751083, 7.16260864929256264513932708125, 7.87747048949143080415613415559, 8.532258564263858957899227098985, 9.452834439660390204373402312738

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