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Properties

Label 2-2100-21.5-c1-0-33
Degree $2$
Conductor $2100$
Sign $0.641 + 0.767i$
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  + (−0.747 + 1.56i)3-s + (0.786 − 2.52i)7-s + (−1.88 − 2.33i)9-s + (−2.34 + 1.35i)11-s − 1.12i·13-s + (3.69 + 6.39i)17-s + (−0.412 − 0.238i)19-s + (3.35 + 3.11i)21-s + (−4.84 − 2.79i)23-s + (5.05 − 1.19i)27-s − 2.20i·29-s + (−2.07 + 1.19i)31-s + (−0.362 − 4.67i)33-s + (4.34 − 7.53i)37-s + (1.75 + 0.840i)39-s + ⋯
L(s)  = 1  + (−0.431 + 0.902i)3-s + (0.297 − 0.954i)7-s + (−0.627 − 0.778i)9-s + (−0.706 + 0.408i)11-s − 0.311i·13-s + (0.895 + 1.55i)17-s + (−0.0945 − 0.0546i)19-s + (0.732 + 0.680i)21-s + (−1.01 − 0.583i)23-s + (0.973 − 0.230i)27-s − 0.409i·29-s + (−0.372 + 0.214i)31-s + (−0.0631 − 0.813i)33-s + (0.714 − 1.23i)37-s + (0.281 + 0.134i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.641 + 0.767i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.641 + 0.767i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2100\)    =    \(2^{2} \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $0.641 + 0.767i$
Analytic conductor: \(16.7685\)
Root analytic conductor: \(4.09494\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2100} (1601, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2100,\ (\ :1/2),\ 0.641 + 0.767i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.054915295\)
\(L(\frac12)\) \(\approx\) \(1.054915295\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.747 - 1.56i)T \)
5 \( 1 \)
7 \( 1 + (-0.786 + 2.52i)T \)
good11 \( 1 + (2.34 - 1.35i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + 1.12iT - 13T^{2} \)
17 \( 1 + (-3.69 - 6.39i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.412 + 0.238i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (4.84 + 2.79i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 2.20iT - 29T^{2} \)
31 \( 1 + (2.07 - 1.19i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-4.34 + 7.53i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 5.42T + 41T^{2} \)
43 \( 1 - 4.16T + 43T^{2} \)
47 \( 1 + (-6.21 + 10.7i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-4.25 + 2.45i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (1.15 + 2.00i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.26 + 1.30i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (7.34 + 12.7i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 9.89iT - 71T^{2} \)
73 \( 1 + (-6.29 + 3.63i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-3.47 + 6.01i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 11.3T + 83T^{2} \)
89 \( 1 + (3.48 - 6.03i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 8.79iT - 97T^{2} \)
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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

−9.105180394527852959305372771766, −8.118657070423141793225716650434, −7.63518790933754584315072045773, −6.48044098307969866661199517074, −5.73635886753409778325274885029, −4.92902043481472656688263311167, −4.05223693481251195208839538337, −3.49326534408295962302356081268, −2.02674835572195497076915630522, −0.43872799479295756226033383708, 1.12722375999671760392558967675, 2.35087329899392222729181816097, 3.04567304865284156006159175161, 4.60055675907354189966317686846, 5.53019602389780587545785287421, 5.84441777659586134697425834724, 6.94156301998250160904820241571, 7.69252645313186762129993241356, 8.242824931913127526474262275992, 9.128331187120807362953318985386

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