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Properties

Label 2-2100-7.2-c1-0-16
Degree $2$
Conductor $2100$
Sign $0.0633 + 0.997i$
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  + (−0.5 − 0.866i)3-s + (−1.32 − 2.29i)7-s + (−0.499 + 0.866i)9-s + (2.82 + 4.88i)11-s − 13-s + (−0.822 − 1.42i)17-s + (2.32 − 4.02i)19-s + (−1.32 + 2.29i)21-s + (−1 + 1.73i)23-s + 0.999·27-s + 7.29·29-s + (−2 − 3.46i)31-s + (2.82 − 4.88i)33-s + (4.14 − 7.18i)37-s + (0.5 + 0.866i)39-s + ⋯
L(s)  = 1  + (−0.288 − 0.499i)3-s + (−0.499 − 0.866i)7-s + (−0.166 + 0.288i)9-s + (0.851 + 1.47i)11-s − 0.277·13-s + (−0.199 − 0.345i)17-s + (0.532 − 0.923i)19-s + (−0.288 + 0.499i)21-s + (−0.208 + 0.361i)23-s + 0.192·27-s + 1.35·29-s + (−0.359 − 0.622i)31-s + (0.491 − 0.851i)33-s + (0.681 − 1.18i)37-s + (0.0800 + 0.138i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0633 + 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0633 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2100\)    =    \(2^{2} \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $0.0633 + 0.997i$
Analytic conductor: \(16.7685\)
Root analytic conductor: \(4.09494\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2100} (1801, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2100,\ (\ :1/2),\ 0.0633 + 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.339495541\)
\(L(\frac12)\) \(\approx\) \(1.339495541\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.5 + 0.866i)T \)
5 \( 1 \)
7 \( 1 + (1.32 + 2.29i)T \)
good11 \( 1 + (-2.82 - 4.88i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + T + 13T^{2} \)
17 \( 1 + (0.822 + 1.42i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.32 + 4.02i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (1 - 1.73i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 7.29T + 29T^{2} \)
31 \( 1 + (2 + 3.46i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-4.14 + 7.18i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 0.354T + 41T^{2} \)
43 \( 1 - 7.29T + 43T^{2} \)
47 \( 1 + (-1.46 + 2.54i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-1.17 - 2.03i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (7.46 + 12.9i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (4.79 - 8.29i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.96 + 6.87i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 3.29T + 71T^{2} \)
73 \( 1 + (3.14 + 5.44i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-3.32 + 5.75i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 3.64T + 83T^{2} \)
89 \( 1 + (-4.46 + 7.73i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 14.2T + 97T^{2} \)
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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

−9.209456349996633530146232070828, −7.84009308089750201624125349596, −7.23073205710021093108019798378, −6.78484944650826718355510102908, −5.90877919349635996304903761073, −4.72288186550299424367227412206, −4.18736906724693971267957233499, −2.94026737046701829449661264902, −1.81789138497934009409432916330, −0.57703796491136508088799652157, 1.11406034530742830409643396898, 2.71426176273988767989977049604, 3.46076733018253984967630880425, 4.39535912166321207273588233561, 5.47484916501269531750040513105, 6.09515861810615427510324717243, 6.63116022640608874013718619850, 7.950570401855339766628794082452, 8.672813959074233833577463371390, 9.212714856418990130262399223511

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