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Properties

Label 2-2100-35.3-c1-0-1
Degree $2$
Conductor $2100$
Sign $-0.980 + 0.197i$
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  + (0.258 + 0.965i)3-s + (2.36 − 1.18i)7-s + (−0.866 + 0.499i)9-s + (−2.25 + 3.90i)11-s + (−4.37 + 4.37i)13-s + (−4.35 + 1.16i)17-s + (−2.51 − 4.34i)19-s + (1.75 + 1.97i)21-s + (1.44 − 5.40i)23-s + (−0.707 − 0.707i)27-s + (−3.92 − 2.26i)31-s + (−4.35 − 1.16i)33-s + (1.67 + 0.448i)37-s + (−5.35 − 3.09i)39-s − 2.22i·41-s + ⋯
L(s)  = 1  + (0.149 + 0.557i)3-s + (0.894 − 0.447i)7-s + (−0.288 + 0.166i)9-s + (−0.680 + 1.17i)11-s + (−1.21 + 1.21i)13-s + (−1.05 + 0.283i)17-s + (−0.575 − 0.997i)19-s + (0.383 + 0.431i)21-s + (0.301 − 1.12i)23-s + (−0.136 − 0.136i)27-s + (−0.704 − 0.406i)31-s + (−0.758 − 0.203i)33-s + (0.275 + 0.0736i)37-s + (−0.857 − 0.495i)39-s − 0.347i·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.980 + 0.197i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.980 + 0.197i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2100\)    =    \(2^{2} \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $-0.980 + 0.197i$
Analytic conductor: \(16.7685\)
Root analytic conductor: \(4.09494\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2100} (493, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2100,\ (\ :1/2),\ -0.980 + 0.197i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3839573618\)
\(L(\frac12)\) \(\approx\) \(0.3839573618\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.258 - 0.965i)T \)
5 \( 1 \)
7 \( 1 + (-2.36 + 1.18i)T \)
good11 \( 1 + (2.25 - 3.90i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (4.37 - 4.37i)T - 13iT^{2} \)
17 \( 1 + (4.35 - 1.16i)T + (14.7 - 8.5i)T^{2} \)
19 \( 1 + (2.51 + 4.34i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.44 + 5.40i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 - 29T^{2} \)
31 \( 1 + (3.92 + 2.26i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1.67 - 0.448i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + 2.22iT - 41T^{2} \)
43 \( 1 + (7.85 + 7.85i)T + 43iT^{2} \)
47 \( 1 + (3.33 - 12.4i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (11.8 - 3.17i)T + (45.8 - 26.5i)T^{2} \)
59 \( 1 + (1.11 - 1.92i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-6.34 + 3.66i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.29 - 4.84i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 - 14.3T + 71T^{2} \)
73 \( 1 + (-1.12 - 4.19i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (1.58 - 0.914i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (11.2 - 11.2i)T - 83iT^{2} \)
89 \( 1 + (3.90 + 6.76i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (3.53 + 3.53i)T + 97iT^{2} \)
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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

−9.525392519220249591837378083715, −8.813823912706370456972694184215, −8.007406052362404154599035384371, −7.12363043045377679403484862351, −6.64832198379212490628067750891, −5.14497567453249581424274259616, −4.59617604212434688115316487870, −4.18962050362632369549625513850, −2.53921365159430505725225215668, −1.95243610191858158889505143132, 0.11936332793645784877664562919, 1.66679498378904363258962505189, 2.63669971002524555076434447405, 3.49398806225215161610301838820, 4.95658661538547454779070375287, 5.39272525391715312910223592109, 6.29616852211731106989665005104, 7.30501802976155754507234917857, 8.128455213480508456786109018836, 8.315565496891621294388416648495

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