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Properties

Label 2-2100-15.8-c1-0-17
Degree $2$
Conductor $2100$
Sign $0.937 - 0.348i$
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  + (1.73 − 0.0412i)3-s + (−0.707 + 0.707i)7-s + (2.99 − 0.142i)9-s + 1.76i·11-s + (0.719 + 0.719i)13-s + (1.55 + 1.55i)17-s − 5.12i·19-s + (−1.19 + 1.25i)21-s + (1.47 − 1.47i)23-s + (5.18 − 0.371i)27-s + 7.63·29-s − 0.104·31-s + (0.0728 + 3.05i)33-s + (0.0126 − 0.0126i)37-s + (1.27 + 1.21i)39-s + ⋯
L(s)  = 1  + (0.999 − 0.0238i)3-s + (−0.267 + 0.267i)7-s + (0.998 − 0.0476i)9-s + 0.532i·11-s + (0.199 + 0.199i)13-s + (0.376 + 0.376i)17-s − 1.17i·19-s + (−0.260 + 0.273i)21-s + (0.306 − 0.306i)23-s + (0.997 − 0.0714i)27-s + 1.41·29-s − 0.0187·31-s + (0.0126 + 0.532i)33-s + (0.00207 − 0.00207i)37-s + (0.204 + 0.194i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.937 - 0.348i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.937 - 0.348i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2100\)    =    \(2^{2} \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $0.937 - 0.348i$
Analytic conductor: \(16.7685\)
Root analytic conductor: \(4.09494\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2100} (1793, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2100,\ (\ :1/2),\ 0.937 - 0.348i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.660266747\)
\(L(\frac12)\) \(\approx\) \(2.660266747\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.73 + 0.0412i)T \)
5 \( 1 \)
7 \( 1 + (0.707 - 0.707i)T \)
good11 \( 1 - 1.76iT - 11T^{2} \)
13 \( 1 + (-0.719 - 0.719i)T + 13iT^{2} \)
17 \( 1 + (-1.55 - 1.55i)T + 17iT^{2} \)
19 \( 1 + 5.12iT - 19T^{2} \)
23 \( 1 + (-1.47 + 1.47i)T - 23iT^{2} \)
29 \( 1 - 7.63T + 29T^{2} \)
31 \( 1 + 0.104T + 31T^{2} \)
37 \( 1 + (-0.0126 + 0.0126i)T - 37iT^{2} \)
41 \( 1 - 9.35iT - 41T^{2} \)
43 \( 1 + (-2.66 - 2.66i)T + 43iT^{2} \)
47 \( 1 + (-3.98 - 3.98i)T + 47iT^{2} \)
53 \( 1 + (5.44 - 5.44i)T - 53iT^{2} \)
59 \( 1 - 5.32T + 59T^{2} \)
61 \( 1 + 4.82T + 61T^{2} \)
67 \( 1 + (-3.01 + 3.01i)T - 67iT^{2} \)
71 \( 1 - 5.20iT - 71T^{2} \)
73 \( 1 + (-7.10 - 7.10i)T + 73iT^{2} \)
79 \( 1 + 15.4iT - 79T^{2} \)
83 \( 1 + (3.76 - 3.76i)T - 83iT^{2} \)
89 \( 1 - 2.98T + 89T^{2} \)
97 \( 1 + (6.41 - 6.41i)T - 97iT^{2} \)
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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

−9.124301519348798330514986291271, −8.434915138782537113659794862935, −7.69995654207660720816903638159, −6.87306181651543701043372466850, −6.20898063246403210034689445766, −4.89454189761391277195676811649, −4.25960035863377194601154741765, −3.11632376093249488320948437214, −2.47302663645573011916972323847, −1.21482401942918115031202620638, 0.997848942328209302701693126103, 2.26565359971348360772224418291, 3.31364363871163639472669863083, 3.85133214274232032164989938868, 4.97089572483714797493508913588, 5.95139427069986422594652507048, 6.88218723751996146787644469748, 7.62194843814141394249999251877, 8.347913158600818265502036462362, 8.932239429644597188451814701202

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