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Properties

Label 2-2100-35.4-c1-0-12
Degree $2$
Conductor $2100$
Sign $0.867 - 0.497i$
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 + (2.29 − 1.32i)7-s + (0.499 + 0.866i)9-s + (0.822 − 1.42i)11-s + 2.64i·13-s + (1.42 + 0.822i)17-s + (4.14 + 7.18i)19-s + 2.64·21-s + (−1.42 + 0.822i)23-s + 0.999i·27-s − 7.64·29-s + (2.14 − 3.71i)31-s + (1.42 − 0.822i)33-s + (0.559 − 0.322i)37-s + (−1.32 + 2.29i)39-s + ⋯
L(s)  = 1  + (0.499 + 0.288i)3-s + (0.866 − 0.499i)7-s + (0.166 + 0.288i)9-s + (0.248 − 0.429i)11-s + 0.733i·13-s + (0.345 + 0.199i)17-s + (0.951 + 1.64i)19-s + 0.577·21-s + (−0.297 + 0.171i)23-s + 0.192i·27-s − 1.41·29-s + (0.385 − 0.667i)31-s + (0.248 − 0.143i)33-s + (0.0919 − 0.0530i)37-s + (−0.211 + 0.366i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.867 - 0.497i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.867 - 0.497i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2100\)    =    \(2^{2} \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $0.867 - 0.497i$
Analytic conductor: \(16.7685\)
Root analytic conductor: \(4.09494\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2100} (949, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2100,\ (\ :1/2),\ 0.867 - 0.497i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.464627084\)
\(L(\frac12)\) \(\approx\) \(2.464627084\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.29 + 1.32i)T \)
good11 \( 1 + (-0.822 + 1.42i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 2.64iT - 13T^{2} \)
17 \( 1 + (-1.42 - 0.822i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-4.14 - 7.18i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.42 - 0.822i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 7.64T + 29T^{2} \)
31 \( 1 + (-2.14 + 3.71i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-0.559 + 0.322i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 - 4.93T + 41T^{2} \)
43 \( 1 - 5.93iT - 43T^{2} \)
47 \( 1 + (-5.19 + 3i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-2.85 - 1.64i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-5.46 + 9.47i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (4 + 6.92i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.559 + 0.322i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 13.6T + 71T^{2} \)
73 \( 1 + (-11.4 - 6.61i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-1.14 - 1.98i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 10.9iT - 83T^{2} \)
89 \( 1 + (7.11 + 12.3i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 8iT - 97T^{2} \)
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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

−9.336762317057215946354425910208, −8.076531789626152356754365945398, −7.972245521374626714118493798596, −6.97642457261687912906154207558, −5.89936144584199752978369878824, −5.15971185832284082968084179178, −4.00582997157017645709586839022, −3.65415584690825586222480666675, −2.20371678573352421096129027996, −1.23964563035760975422167316730, 0.972353287839318002328325777768, 2.19250757426410375564997484029, 2.99266769682770526349117020715, 4.13603771521462577863771038356, 5.10694073327198121157577927019, 5.72616521314398311649549114182, 6.96578780129991949175109346568, 7.50108501577342212390425435792, 8.243995396094624116419191817729, 9.054226634452799674863661761931

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