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Properties

Label 2-2100-35.9-c1-0-20
Degree $2$
Conductor $2100$
Sign $-0.0667 + 0.997i$
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  + (0.866 − 0.5i)3-s + (1.73 − 2i)7-s + (0.499 − 0.866i)9-s + (−1 − 1.73i)11-s − 2i·13-s + (−1.73 + i)17-s + (2 − 3.46i)19-s + (0.499 − 2.59i)21-s + (6.92 + 4i)23-s − 0.999i·27-s − 4·29-s + (−1.5 − 2.59i)31-s + (−1.73 − 0.999i)33-s + (−7.79 − 4.5i)37-s + (−1 − 1.73i)39-s + ⋯
L(s)  = 1  + (0.499 − 0.288i)3-s + (0.654 − 0.755i)7-s + (0.166 − 0.288i)9-s + (−0.301 − 0.522i)11-s − 0.554i·13-s + (−0.420 + 0.242i)17-s + (0.458 − 0.794i)19-s + (0.109 − 0.566i)21-s + (1.44 + 0.834i)23-s − 0.192i·27-s − 0.742·29-s + (−0.269 − 0.466i)31-s + (−0.301 − 0.174i)33-s + (−1.28 − 0.739i)37-s + (−0.160 − 0.277i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0667 + 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0667 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2100\)    =    \(2^{2} \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $-0.0667 + 0.997i$
Analytic conductor: \(16.7685\)
Root analytic conductor: \(4.09494\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2100} (1549, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2100,\ (\ :1/2),\ -0.0667 + 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.053278706\)
\(L(\frac12)\) \(\approx\) \(2.053278706\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.73 + 2i)T \)
good11 \( 1 + (1 + 1.73i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 2iT - 13T^{2} \)
17 \( 1 + (1.73 - i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2 + 3.46i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-6.92 - 4i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 4T + 29T^{2} \)
31 \( 1 + (1.5 + 2.59i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (7.79 + 4.5i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 - 6T + 41T^{2} \)
43 \( 1 + iT - 43T^{2} \)
47 \( 1 + (5.19 + 3i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-1.73 + i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (3 + 5.19i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.5 + 0.866i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (10.3 - 6i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 10T + 71T^{2} \)
73 \( 1 + (-0.866 + 0.5i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-3.5 + 6.06i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 18iT - 83T^{2} \)
89 \( 1 + (-5 + 8.66i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 5iT - 97T^{2} \)
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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

−8.875032723608722782075468199645, −8.048490285029435652777958966019, −7.40656531612587841301862979050, −6.82732653031569884135523076577, −5.61097645566576841973699276621, −4.92437796143914723075494856445, −3.82243157560705699673072942615, −3.05588597465297942748305154182, −1.87068492828571994887887238356, −0.68572778958977538186568091797, 1.55706665109378595730675548313, 2.47824314985217699343970059238, 3.43650300761406024814772363261, 4.62331873963495968221215442833, 5.07580866531011820359701447742, 6.13279768253071941751462265991, 7.13227646087053838409239502981, 7.79871015220028981909791343952, 8.733914744925270928750475552803, 9.095230914604287461112656753148

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