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Properties

Label 2-2100-21.17-c1-0-41
Degree $2$
Conductor $2100$
Sign $-0.0867 + 0.996i$
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  + (1.66 − 0.478i)3-s + (−1.25 + 2.33i)7-s + (2.54 − 1.59i)9-s + (−4.63 − 2.67i)11-s − 4.13i·13-s + (0.0773 − 0.134i)17-s + (−3.40 + 1.96i)19-s + (−0.971 + 4.47i)21-s + (5.19 − 2.99i)23-s + (3.46 − 3.86i)27-s − 10.3i·29-s + (6.70 + 3.86i)31-s + (−9.00 − 2.23i)33-s + (−5.24 − 9.07i)37-s + (−1.97 − 6.88i)39-s + ⋯
L(s)  = 1  + (0.961 − 0.276i)3-s + (−0.473 + 0.880i)7-s + (0.847 − 0.531i)9-s + (−1.39 − 0.807i)11-s − 1.14i·13-s + (0.0187 − 0.0325i)17-s + (−0.781 + 0.450i)19-s + (−0.211 + 0.977i)21-s + (1.08 − 0.625i)23-s + (0.667 − 0.744i)27-s − 1.91i·29-s + (1.20 + 0.694i)31-s + (−1.56 − 0.389i)33-s + (−0.861 − 1.49i)37-s + (−0.317 − 1.10i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0867 + 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0867 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2100\)    =    \(2^{2} \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $-0.0867 + 0.996i$
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: \((2,\ 2100,\ (\ :1/2),\ -0.0867 + 0.996i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.763053450\)
\(L(\frac12)\) \(\approx\) \(1.763053450\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.66 + 0.478i)T \)
5 \( 1 \)
7 \( 1 + (1.25 - 2.33i)T \)
good11 \( 1 + (4.63 + 2.67i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + 4.13iT - 13T^{2} \)
17 \( 1 + (-0.0773 + 0.134i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.40 - 1.96i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-5.19 + 2.99i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 10.3iT - 29T^{2} \)
31 \( 1 + (-6.70 - 3.86i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (5.24 + 9.07i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 2.33T + 41T^{2} \)
43 \( 1 - 1.78T + 43T^{2} \)
47 \( 1 + (1.80 + 3.11i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (4.29 + 2.47i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-5.27 + 9.12i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (3.25 - 1.87i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.444 + 0.770i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 11.6iT - 71T^{2} \)
73 \( 1 + (10.6 + 6.12i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-3.61 - 6.25i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 5.14T + 83T^{2} \)
89 \( 1 + (-8.58 - 14.8i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 4.28iT - 97T^{2} \)
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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

−8.614525960193278925637282863288, −8.292896898554958370117475993516, −7.60326211216175255486299822664, −6.52615736846503795828167897156, −5.78798549772411506727305769888, −4.94392782133296225644781692349, −3.66349529946405676041570861947, −2.80754332498920063273104521714, −2.31814606964360548547849293245, −0.52888838200155968638389450343, 1.50667068376092738688077004062, 2.64529062080018590227625464803, 3.40676575430584075936774191712, 4.55825747993808991654460984677, 4.86575789096598682126530338052, 6.42738117253006925522163239022, 7.19875128200106981813090932395, 7.63038660449683318740413957991, 8.668266180330729519196553542374, 9.232146544465187371960673187869

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