Properties

Label 2-2100-21.17-c1-0-47
Degree $2$
Conductor $2100$
Sign $-0.999 + 0.0261i$
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  + (−0.546 − 1.64i)3-s + (1.08 − 2.41i)7-s + (−2.40 + 1.79i)9-s + (−1.17 − 0.675i)11-s − 4.94i·13-s + (2.87 − 4.97i)17-s + (2.84 − 1.64i)19-s + (−4.55 − 0.460i)21-s + (−4.33 + 2.50i)23-s + (4.26 + 2.96i)27-s + 5.68i·29-s + (−2.45 − 1.41i)31-s + (−0.471 + 2.29i)33-s + (1.92 + 3.33i)37-s + (−8.12 + 2.69i)39-s + ⋯
L(s)  = 1  + (−0.315 − 0.948i)3-s + (0.409 − 0.912i)7-s + (−0.801 + 0.598i)9-s + (−0.353 − 0.203i)11-s − 1.37i·13-s + (0.696 − 1.20i)17-s + (0.653 − 0.377i)19-s + (−0.994 − 0.100i)21-s + (−0.903 + 0.521i)23-s + (0.820 + 0.571i)27-s + 1.05i·29-s + (−0.440 − 0.254i)31-s + (−0.0820 + 0.399i)33-s + (0.316 + 0.548i)37-s + (−1.30 + 0.432i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0261i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 + 0.0261i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2100\)    =    \(2^{2} \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $-0.999 + 0.0261i$
Analytic conductor: \(16.7685\)
Root analytic conductor: \(4.09494\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2100} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2100,\ (\ :1/2),\ -0.999 + 0.0261i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.096927386\)
\(L(\frac12)\) \(\approx\) \(1.096927386\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.546 + 1.64i)T \)
5 \( 1 \)
7 \( 1 + (-1.08 + 2.41i)T \)
good11 \( 1 + (1.17 + 0.675i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + 4.94iT - 13T^{2} \)
17 \( 1 + (-2.87 + 4.97i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.84 + 1.64i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (4.33 - 2.50i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 5.68iT - 29T^{2} \)
31 \( 1 + (2.45 + 1.41i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1.92 - 3.33i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 3.73T + 41T^{2} \)
43 \( 1 + 4.06T + 43T^{2} \)
47 \( 1 + (2.84 + 4.92i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-1.26 - 0.730i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-4.34 + 7.52i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.65 + 0.954i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.51 + 4.36i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 3.38iT - 71T^{2} \)
73 \( 1 + (14.1 + 8.16i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (2.41 + 4.18i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 16.6T + 83T^{2} \)
89 \( 1 + (-8.08 - 13.9i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 12.0iT - 97T^{2} \)
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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

−8.413551260222198504929628399120, −7.70445435451149274405023041606, −7.40028926873353387589457500194, −6.46521024683273345494290198451, −5.38431788323282882967341134323, −5.06392656392535819159612812501, −3.55794289067514503543957157538, −2.73496289531707022954666443509, −1.37076565854177648256726476934, −0.41706881291630047972420163418, 1.69907149201159378828890180210, 2.79730414899528537109098813651, 4.00542774870664086160288138781, 4.51825803873104768846407742592, 5.75040030690243851661706610816, 5.88113377257010233439111354193, 7.13093191916392343338776107076, 8.244407423807733186891771599110, 8.678465783856670209366047112352, 9.686014173757127419987104499064

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