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Properties

Label 2-2100-21.17-c1-0-4
Degree $2$
Conductor $2100$
Sign $-0.230 - 0.973i$
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  + (1.25 − 1.19i)3-s + (−1.60 + 2.10i)7-s + (0.157 − 2.99i)9-s + (2.05 + 1.18i)11-s + 0.748i·13-s + (−3.77 + 6.53i)17-s + (−6.11 + 3.53i)19-s + (0.485 + 4.55i)21-s + (−2.83 + 1.63i)23-s + (−3.37 − 3.95i)27-s + 2.48i·29-s + (−6.84 − 3.95i)31-s + (4.00 − 0.961i)33-s + (2.15 + 3.73i)37-s + (0.892 + 0.940i)39-s + ⋯
L(s)  = 1  + (0.725 − 0.688i)3-s + (−0.607 + 0.794i)7-s + (0.0523 − 0.998i)9-s + (0.620 + 0.358i)11-s + 0.207i·13-s + (−0.914 + 1.58i)17-s + (−1.40 + 0.810i)19-s + (0.105 + 0.994i)21-s + (−0.590 + 0.340i)23-s + (−0.649 − 0.760i)27-s + 0.461i·29-s + (−1.22 − 0.709i)31-s + (0.696 − 0.167i)33-s + (0.354 + 0.614i)37-s + (0.142 + 0.150i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.230 - 0.973i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.230 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2100\)    =    \(2^{2} \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $-0.230 - 0.973i$
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: \((2,\ 2100,\ (\ :1/2),\ -0.230 - 0.973i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.073792012\)
\(L(\frac12)\) \(\approx\) \(1.073792012\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.25 + 1.19i)T \)
5 \( 1 \)
7 \( 1 + (1.60 - 2.10i)T \)
good11 \( 1 + (-2.05 - 1.18i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 - 0.748iT - 13T^{2} \)
17 \( 1 + (3.77 - 6.53i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (6.11 - 3.53i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (2.83 - 1.63i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 2.48iT - 29T^{2} \)
31 \( 1 + (6.84 + 3.95i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-2.15 - 3.73i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 10.8T + 41T^{2} \)
43 \( 1 - 3.03T + 43T^{2} \)
47 \( 1 + (-3.22 - 5.59i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (0.0935 + 0.0540i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-6.60 + 11.4i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-6.90 + 3.98i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.94 + 5.09i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 13.9iT - 71T^{2} \)
73 \( 1 + (-1.35 - 0.780i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (1.27 + 2.21i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 0.901T + 83T^{2} \)
89 \( 1 + (-2.43 - 4.21i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 12.9iT - 97T^{2} \)
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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

−9.202942499037503799973453680270, −8.464611946159383581721240706370, −8.036292271402488853807092391880, −6.72980557638199434958047011290, −6.47439279272492876464480490998, −5.60771945292409969907746169222, −4.10630107212317978234318543921, −3.60603345388212805651493569193, −2.25097299891011895541942925912, −1.73101509046596379099111390904, 0.31212192618537857961906157540, 2.13590031087996033628166289171, 3.05872009304227255278239301561, 4.00143491088767162800948921251, 4.53965608434334481826921088963, 5.61344385051104440958481047945, 6.82211028045335217811984979259, 7.16640729491043633293265636306, 8.356277718184682761171512628262, 8.941452105911276863272466126979

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