Properties

Label 2-2100-105.59-c1-0-47
Degree $2$
Conductor $2100$
Sign $-0.844 + 0.535i$
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 + 0.990i)3-s + (0.156 − 2.64i)7-s + (1.03 + 2.81i)9-s + (−4.92 − 2.84i)11-s − 1.43·13-s + (−2.81 − 1.62i)17-s + (−5.43 + 3.13i)19-s + (2.83 − 3.59i)21-s + (−0.510 − 0.884i)23-s + (−1.31 + 5.02i)27-s − 4.95i·29-s + (−2.28 − 1.31i)31-s + (−4.17 − 8.91i)33-s + (−7.88 + 4.55i)37-s + (−2.04 − 1.42i)39-s + ⋯
L(s)  = 1  + (0.820 + 0.572i)3-s + (0.0592 − 0.998i)7-s + (0.345 + 0.938i)9-s + (−1.48 − 0.857i)11-s − 0.398·13-s + (−0.682 − 0.393i)17-s + (−1.24 + 0.719i)19-s + (0.619 − 0.784i)21-s + (−0.106 − 0.184i)23-s + (−0.253 + 0.967i)27-s − 0.920i·29-s + (−0.410 − 0.236i)31-s + (−0.727 − 1.55i)33-s + (−1.29 + 0.748i)37-s + (−0.326 − 0.227i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.844 + 0.535i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.844 + 0.535i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2100\)    =    \(2^{2} \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $-0.844 + 0.535i$
Analytic conductor: \(16.7685\)
Root analytic conductor: \(4.09494\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2100} (1949, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2100,\ (\ :1/2),\ -0.844 + 0.535i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4098479591\)
\(L(\frac12)\) \(\approx\) \(0.4098479591\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.42 - 0.990i)T \)
5 \( 1 \)
7 \( 1 + (-0.156 + 2.64i)T \)
good11 \( 1 + (4.92 + 2.84i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + 1.43T + 13T^{2} \)
17 \( 1 + (2.81 + 1.62i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (5.43 - 3.13i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.510 + 0.884i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 4.95iT - 29T^{2} \)
31 \( 1 + (2.28 + 1.31i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (7.88 - 4.55i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + 0.203T + 41T^{2} \)
43 \( 1 - 3.91iT - 43T^{2} \)
47 \( 1 + (9.98 - 5.76i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-5.32 + 9.21i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-0.739 + 1.28i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-7.62 + 4.40i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-4.05 - 2.34i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 1.08iT - 71T^{2} \)
73 \( 1 + (4.51 - 7.82i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-5.80 - 10.0i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 10.1iT - 83T^{2} \)
89 \( 1 + (7.53 + 13.0i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 10.5T + 97T^{2} \)
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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

−8.435757655931699449558768104067, −8.296484676169297217521353731975, −7.42311753892956420010303566313, −6.55148480865169554212448040636, −5.38419117448986188797374646015, −4.61327232985066654928861228073, −3.81998533657572436071330344125, −2.91203731562339312538101864548, −1.98077582127483658076708412257, −0.11230276602312145122017198987, 2.01839111127976349860748169897, 2.33901440469346184709631624017, 3.43353058220888420171547814384, 4.66256797160849041956454343951, 5.38646195577402070220359576729, 6.47858064734889875077383884077, 7.17925772700889904084513221364, 7.905770035083445688246246843846, 8.770705018323474491046645645332, 9.040051887847311827081150566498

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