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Properties

Label 2-2100-105.59-c1-0-43
Degree $2$
Conductor $2100$
Sign $-0.929 + 0.367i$
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 + (0.866 − 2.5i)7-s + (−1.5 − 2.59i)9-s − 5.19·13-s + (7.5 − 4.33i)19-s + (−3 − 3.46i)21-s − 5.19·27-s + (−1.5 − 0.866i)31-s + (−9.52 + 5.5i)37-s + (−4.5 + 7.79i)39-s − 13i·43-s + (−5.5 − 4.33i)49-s − 15i·57-s + (−6 + 3.46i)61-s + (−7.79 + 1.5i)63-s + ⋯
L(s)  = 1  + (0.499 − 0.866i)3-s + (0.327 − 0.944i)7-s + (−0.5 − 0.866i)9-s − 1.44·13-s + (1.72 − 0.993i)19-s + (−0.654 − 0.755i)21-s − 1.00·27-s + (−0.269 − 0.155i)31-s + (−1.56 + 0.904i)37-s + (−0.720 + 1.24i)39-s − 1.98i·43-s + (−0.785 − 0.618i)49-s − 1.98i·57-s + (−0.768 + 0.443i)61-s + (−0.981 + 0.188i)63-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.929 + 0.367i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.929 + 0.367i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2100\)    =    \(2^{2} \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $-0.929 + 0.367i$
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: \((2,\ 2100,\ (\ :1/2),\ -0.929 + 0.367i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.512913501\)
\(L(\frac12)\) \(\approx\) \(1.512913501\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.866 + 2.5i)T \)
good11 \( 1 + (5.5 + 9.52i)T^{2} \)
13 \( 1 + 5.19T + 13T^{2} \)
17 \( 1 + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-7.5 + 4.33i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 29T^{2} \)
31 \( 1 + (1.5 + 0.866i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (9.52 - 5.5i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 + 13iT - 43T^{2} \)
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 \( 1 + (6 - 3.46i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (4.33 + 2.5i)T + (33.5 + 58.0i)T^{2} \)
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 \( 1 - 13.8T + 97T^{2} \)
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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

−8.687709039675652397257019868662, −7.78571896353337180998830103442, −7.16691653752784696541743586707, −6.88551286772474603474186039733, −5.51089910090811474324218222998, −4.79018757066439254313120474280, −3.61220258606196135197350198560, −2.76451854517226619934096909615, −1.66355037858585246827276961960, −0.47593651905714215578408868049, 1.81568340986610606066463242446, 2.80658142294612919484024782511, 3.56212983273968136950541809069, 4.81346295720673767294983912359, 5.21199448030299737853984056248, 6.08214105373282954791486049923, 7.47493148577492415888661715864, 7.85248994584560436268628043355, 8.874728211475526168322732006971, 9.415764913984791010494724936732

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