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Properties

Label 2-2100-35.4-c1-0-23
Degree $2$
Conductor $2100$
Sign $-0.705 + 0.708i$
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  + (0.866 + 0.5i)3-s + (0.358 − 2.62i)7-s + (0.499 + 0.866i)9-s + (−2.12 + 3.67i)11-s − 5i·13-s + (−3.67 − 2.12i)17-s + (−2.62 − 4.54i)19-s + (1.62 − 2.09i)21-s + (−5.19 + 3i)23-s + 0.999i·27-s − 8.48·29-s + (2 − 3.46i)31-s + (−3.67 + 2.12i)33-s + (4.33 − 2.5i)37-s + (2.5 − 4.33i)39-s + ⋯
L(s)  = 1  + (0.499 + 0.288i)3-s + (0.135 − 0.990i)7-s + (0.166 + 0.288i)9-s + (−0.639 + 1.10i)11-s − 1.38i·13-s + (−0.891 − 0.514i)17-s + (−0.601 − 1.04i)19-s + (0.353 − 0.456i)21-s + (−1.08 + 0.625i)23-s + 0.192i·27-s − 1.57·29-s + (0.359 − 0.622i)31-s + (−0.639 + 0.369i)33-s + (0.711 − 0.410i)37-s + (0.400 − 0.693i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.705 + 0.708i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.705 + 0.708i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2100\)    =    \(2^{2} \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $-0.705 + 0.708i$
Analytic conductor: \(16.7685\)
Root analytic conductor: \(4.09494\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2100} (949, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2100,\ (\ :1/2),\ -0.705 + 0.708i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8266878402\)
\(L(\frac12)\) \(\approx\) \(0.8266878402\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.866 - 0.5i)T \)
5 \( 1 \)
7 \( 1 + (-0.358 + 2.62i)T \)
good11 \( 1 + (2.12 - 3.67i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 5iT - 13T^{2} \)
17 \( 1 + (3.67 + 2.12i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.62 + 4.54i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (5.19 - 3i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 8.48T + 29T^{2} \)
31 \( 1 + (-2 + 3.46i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-4.33 + 2.5i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + 10.2T + 41T^{2} \)
43 \( 1 - 12.4iT - 43T^{2} \)
47 \( 1 + (-8.87 + 5.12i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-3.67 - 2.12i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-5.12 + 8.87i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.5 - 0.866i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (10.1 + 5.86i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 8.48T + 71T^{2} \)
73 \( 1 + (3.01 + 1.74i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-4.62 - 8.00i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 1.75iT - 83T^{2} \)
89 \( 1 + (-3.87 - 6.71i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 17.9iT - 97T^{2} \)
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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

−8.840699578698769405387894346271, −7.77788343887934996947445172827, −7.57275170921708753363150980619, −6.62363801597879619649555381686, −5.45990458941283721461256413218, −4.63227674853866442180447024427, −3.96281042165988638502827969699, −2.86125004635927132381057750766, −1.93500097635828772428695326287, −0.24484853206573739426137439709, 1.79009295263823960500742932373, 2.40653219996852563387113829125, 3.61005810325108284499131558967, 4.40768554588423201530090168853, 5.66547243544205250220429951766, 6.15491311362189356850726081728, 7.06499427289959810660356682276, 8.076308074128384925120985616884, 8.704998903088712744392828671528, 9.001727225679917429697103859975

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