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Properties

Label 2-2100-21.17-c1-0-43
Degree $2$
Conductor $2100$
Sign $-0.273 + 0.961i$
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  + (0.956 − 1.44i)3-s + (2.60 − 0.477i)7-s + (−1.16 − 2.76i)9-s + (1.34 + 0.773i)11-s − 4.18i·13-s + (2.55 − 4.42i)17-s + (−4.62 + 2.67i)19-s + (1.80 − 4.21i)21-s + (−3.15 + 1.82i)23-s + (−5.10 − 0.954i)27-s − 9.79i·29-s + (6.79 + 3.92i)31-s + (2.39 − 1.19i)33-s + (1.71 + 2.96i)37-s + (−6.04 − 4.00i)39-s + ⋯
L(s)  = 1  + (0.552 − 0.833i)3-s + (0.983 − 0.180i)7-s + (−0.389 − 0.920i)9-s + (0.404 + 0.233i)11-s − 1.16i·13-s + (0.619 − 1.07i)17-s + (−1.06 + 0.613i)19-s + (0.392 − 0.919i)21-s + (−0.658 + 0.380i)23-s + (−0.982 − 0.183i)27-s − 1.81i·29-s + (1.22 + 0.705i)31-s + (0.417 − 0.208i)33-s + (0.281 + 0.488i)37-s + (−0.967 − 0.641i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.273 + 0.961i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.273 + 0.961i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2100\)    =    \(2^{2} \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $-0.273 + 0.961i$
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: \((2,\ 2100,\ (\ :1/2),\ -0.273 + 0.961i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.257607534\)
\(L(\frac12)\) \(\approx\) \(2.257607534\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.956 + 1.44i)T \)
5 \( 1 \)
7 \( 1 + (-2.60 + 0.477i)T \)
good11 \( 1 + (-1.34 - 0.773i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + 4.18iT - 13T^{2} \)
17 \( 1 + (-2.55 + 4.42i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (4.62 - 2.67i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (3.15 - 1.82i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 9.79iT - 29T^{2} \)
31 \( 1 + (-6.79 - 3.92i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1.71 - 2.96i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 8.82T + 41T^{2} \)
43 \( 1 - 3.79T + 43T^{2} \)
47 \( 1 + (-1.24 - 2.16i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-0.684 - 0.395i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-2.73 + 4.73i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (6.76 - 3.90i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-5.58 + 9.67i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 7.97iT - 71T^{2} \)
73 \( 1 + (-10.7 - 6.19i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-1.12 - 1.94i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 5.26T + 83T^{2} \)
89 \( 1 + (7.65 + 13.2i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 15.9iT - 97T^{2} \)
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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

−8.565713054407043445087900229936, −8.017780287417440388400653447449, −7.59574313161713740658201662126, −6.58643845077686328031774014642, −5.83438653254236880760564764912, −4.83919171959186898733303468203, −3.84741257161573831046974339310, −2.80643714687744983081723687639, −1.85045239417783377126914173160, −0.75470352214213315819163044801, 1.60348743370820589863201234218, 2.49728034415149156509040858890, 3.78822212725364750896475550366, 4.34660723158027512873621454112, 5.14713874133240495375719181067, 6.12089417710344036900293113960, 7.04404788133480083183052436618, 8.142186740967712984923081197416, 8.566273932710746955514456548075, 9.164548164816649381067894415921

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