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Properties

Label 2-2100-105.89-c1-0-10
Degree $2$
Conductor $2100$
Sign $0.338 - 0.941i$
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  + (−1.68 + 0.404i)3-s + (2.10 − 1.60i)7-s + (2.67 − 1.36i)9-s + (−2.05 + 1.18i)11-s − 0.748·13-s + (−6.53 + 3.77i)17-s + (6.11 + 3.53i)19-s + (−2.88 + 3.55i)21-s + (1.63 − 2.83i)23-s + (−3.95 + 3.37i)27-s − 2.48i·29-s + (−6.84 + 3.95i)31-s + (2.98 − 2.83i)33-s + (3.73 + 2.15i)37-s + (1.26 − 0.302i)39-s + ⋯
L(s)  = 1  + (−0.972 + 0.233i)3-s + (0.794 − 0.607i)7-s + (0.891 − 0.453i)9-s + (−0.620 + 0.358i)11-s − 0.207·13-s + (−1.58 + 0.914i)17-s + (1.40 + 0.810i)19-s + (−0.630 + 0.776i)21-s + (0.340 − 0.590i)23-s + (−0.760 + 0.649i)27-s − 0.461i·29-s + (−1.22 + 0.709i)31-s + (0.519 − 0.493i)33-s + (0.614 + 0.354i)37-s + (0.201 − 0.0484i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.338 - 0.941i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.338 - 0.941i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2100\)    =    \(2^{2} \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $0.338 - 0.941i$
Analytic conductor: \(16.7685\)
Root analytic conductor: \(4.09494\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2100} (1349, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2100,\ (\ :1/2),\ 0.338 - 0.941i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.053567080\)
\(L(\frac12)\) \(\approx\) \(1.053567080\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.68 - 0.404i)T \)
5 \( 1 \)
7 \( 1 + (-2.10 + 1.60i)T \)
good11 \( 1 + (2.05 - 1.18i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + 0.748T + 13T^{2} \)
17 \( 1 + (6.53 - 3.77i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-6.11 - 3.53i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.63 + 2.83i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 2.48iT - 29T^{2} \)
31 \( 1 + (6.84 - 3.95i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-3.73 - 2.15i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 - 10.8T + 41T^{2} \)
43 \( 1 + 3.03iT - 43T^{2} \)
47 \( 1 + (5.59 + 3.22i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (0.0540 + 0.0935i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-6.60 - 11.4i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-6.90 - 3.98i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (5.09 - 2.94i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 13.9iT - 71T^{2} \)
73 \( 1 + (0.780 + 1.35i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-1.27 + 2.21i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 0.901iT - 83T^{2} \)
89 \( 1 + (-2.43 + 4.21i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 12.9T + 97T^{2} \)
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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

−9.365867226263057253467523014614, −8.443430893808251758624338222982, −7.50956490051265458213171546608, −7.00142767969405927867496937329, −5.98478366086205699518272803958, −5.23024637078387969478182019721, −4.48447233724758304608626323378, −3.79136075258719226612587845547, −2.22688747637137729134185728172, −1.05711833505511963693736251582, 0.50045460962299339770423481468, 1.89746305242540086693945574929, 2.88127828787414178419808823155, 4.37883337433537395340733688983, 5.12355451268179289773815631433, 5.56996051058357680638020272601, 6.60105224138674024867126862881, 7.40804934365032290138306972583, 7.946938369357516506344441875946, 9.188469074779536421523845002727

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