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Properties

Label 2-2100-105.59-c1-0-1
Degree $2$
Conductor $2100$
Sign $-0.254 - 0.967i$
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 − 1.5i)3-s + (−2.59 + 0.5i)7-s + (−1.5 − 2.59i)9-s − 1.73·13-s + (−4.5 + 2.59i)19-s + (−1.5 + 4.33i)21-s − 5.19·27-s + (9 + 5.19i)31-s + (−9.52 + 5.5i)37-s + (−1.49 + 2.59i)39-s + 8i·43-s + (6.5 − 2.59i)49-s + 9i·57-s + (−7.5 + 4.33i)61-s + (5.19 + 6i)63-s + ⋯
L(s)  = 1  + (0.499 − 0.866i)3-s + (−0.981 + 0.188i)7-s + (−0.5 − 0.866i)9-s − 0.480·13-s + (−1.03 + 0.596i)19-s + (−0.327 + 0.944i)21-s − 1.00·27-s + (1.61 + 0.933i)31-s + (−1.56 + 0.904i)37-s + (−0.240 + 0.416i)39-s + 1.21i·43-s + (0.928 − 0.371i)49-s + 1.19i·57-s + (−0.960 + 0.554i)61-s + (0.654 + 0.755i)63-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.254 - 0.967i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.254 - 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2100\)    =    \(2^{2} \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $-0.254 - 0.967i$
Analytic conductor: \(16.7685\)
Root analytic conductor: \(4.09494\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2100} (1949, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2100,\ (\ :1/2),\ -0.254 - 0.967i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4356402802\)
\(L(\frac12)\) \(\approx\) \(0.4356402802\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.866 + 1.5i)T \)
5 \( 1 \)
7 \( 1 + (2.59 - 0.5i)T \)
good11 \( 1 + (5.5 + 9.52i)T^{2} \)
13 \( 1 + 1.73T + 13T^{2} \)
17 \( 1 + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (4.5 - 2.59i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 29T^{2} \)
31 \( 1 + (-9 - 5.19i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (9.52 - 5.5i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 - 8iT - 43T^{2} \)
47 \( 1 + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (7.5 - 4.33i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (9.52 + 5.5i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 + (0.866 - 1.5i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-8.5 - 14.7i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 83T^{2} \)
89 \( 1 + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 19.0T + 97T^{2} \)
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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

−9.223535388893986940269644842120, −8.469099585399344404101176548919, −7.87672195824225073065866693801, −6.76926405654889521700292687127, −6.52524163293491696802384785619, −5.55528232701351724497699340138, −4.35957753746558317742316602531, −3.26338426665225181151839098670, −2.60528663132931696460484937669, −1.42025949416906607433129148542, 0.13663626241023770969549641904, 2.20328068816299253490870850841, 3.03386024321708511224590664998, 3.94611131814498482222822657044, 4.65658897799173012480975950151, 5.64056368436443038121618241452, 6.53589069146491686845538298225, 7.35245303904895148105647505684, 8.282978049675629225701675473500, 9.022875869255888871645795971399

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