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Properties

Label 2-2100-105.68-c0-0-2
Degree $2$
Conductor $2100$
Sign $0.910 - 0.413i$
Analytic cond. $1.04803$
Root an. cond. $1.02373$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.965 + 0.258i)3-s + (0.965 − 0.258i)7-s + (0.866 + 0.499i)9-s + (−0.707 + 0.707i)13-s + (−0.866 + 1.5i)19-s + 21-s + (0.707 + 0.707i)27-s + (1.5 − 0.866i)31-s + (−0.448 − 1.67i)37-s + (−0.866 + 0.500i)39-s + (−1.22 − 1.22i)43-s + (0.866 − 0.499i)49-s + (−1.22 + 1.22i)57-s + (0.965 + 0.258i)63-s + (−1.67 − 0.448i)67-s + ⋯
L(s)  = 1  + (0.965 + 0.258i)3-s + (0.965 − 0.258i)7-s + (0.866 + 0.499i)9-s + (−0.707 + 0.707i)13-s + (−0.866 + 1.5i)19-s + 21-s + (0.707 + 0.707i)27-s + (1.5 − 0.866i)31-s + (−0.448 − 1.67i)37-s + (−0.866 + 0.500i)39-s + (−1.22 − 1.22i)43-s + (0.866 − 0.499i)49-s + (−1.22 + 1.22i)57-s + (0.965 + 0.258i)63-s + (−1.67 − 0.448i)67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.910 - 0.413i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.910 - 0.413i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2100\)    =    \(2^{2} \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $0.910 - 0.413i$
Analytic conductor: \(1.04803\)
Root analytic conductor: \(1.02373\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2100} (593, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2100,\ (\ :0),\ 0.910 - 0.413i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.766055340\)
\(L(\frac12)\) \(\approx\) \(1.766055340\)
\(L(1)\) 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 \( 1 + (-0.965 - 0.258i)T \)
5 \( 1 \)
7 \( 1 + (-0.965 + 0.258i)T \)
good11 \( 1 + (0.5 - 0.866i)T^{2} \)
13 \( 1 + (0.707 - 0.707i)T - iT^{2} \)
17 \( 1 + (0.866 + 0.5i)T^{2} \)
19 \( 1 + (0.866 - 1.5i)T + (-0.5 - 0.866i)T^{2} \)
23 \( 1 + (-0.866 + 0.5i)T^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 + (-1.5 + 0.866i)T + (0.5 - 0.866i)T^{2} \)
37 \( 1 + (0.448 + 1.67i)T + (-0.866 + 0.5i)T^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 + (1.22 + 1.22i)T + iT^{2} \)
47 \( 1 + (-0.866 + 0.5i)T^{2} \)
53 \( 1 + (0.866 + 0.5i)T^{2} \)
59 \( 1 + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (0.5 + 0.866i)T^{2} \)
67 \( 1 + (1.67 + 0.448i)T + (0.866 + 0.5i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + (0.965 + 0.258i)T + (0.866 + 0.5i)T^{2} \)
79 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
83 \( 1 + iT^{2} \)
89 \( 1 + (0.5 + 0.866i)T^{2} \)
97 \( 1 + (-1.41 - 1.41i)T + iT^{2} \)
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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

−9.206487736433943890359698183453, −8.555186686020699933265615916639, −7.85899440264884426271411918653, −7.30521776544419520617471562294, −6.27753049747185965532118923706, −5.13363840062381935359766167564, −4.32969160129989271305096442946, −3.72219632302965796290347689337, −2.38037071433561241916760339144, −1.68580323058487898409502559381, 1.35093327462798419083978319162, 2.50174319384013010363207105439, 3.13449517558234501863118962822, 4.58437513763845363900613186168, 4.88615937348950763466938400600, 6.28630303228383993132731072213, 7.03798931413965310003979527287, 7.86536072808857315033621750856, 8.448415657712571029589290197730, 8.988636256567794050799568312868

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