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Properties

Label 2-2100-105.59-c1-0-17
Degree $2$
Conductor $2100$
Sign $0.572 - 0.819i$
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  + (1.56 + 0.747i)3-s + (2.60 − 0.456i)7-s + (1.88 + 2.33i)9-s + (−0.698 − 0.403i)11-s − 3.86·13-s + (1.83 + 1.05i)17-s + (−2.70 + 1.56i)19-s + (4.41 + 1.23i)21-s + (2.43 + 4.21i)23-s + (1.19 + 5.05i)27-s + 6.67i·29-s + (5.65 + 3.26i)31-s + (−0.789 − 1.15i)33-s + (6.75 − 3.89i)37-s + (−6.04 − 2.89i)39-s + ⋯
L(s)  = 1  + (0.902 + 0.431i)3-s + (0.985 − 0.172i)7-s + (0.627 + 0.778i)9-s + (−0.210 − 0.121i)11-s − 1.07·13-s + (0.445 + 0.257i)17-s + (−0.620 + 0.358i)19-s + (0.962 + 0.269i)21-s + (0.508 + 0.879i)23-s + (0.229 + 0.973i)27-s + 1.23i·29-s + (1.01 + 0.585i)31-s + (−0.137 − 0.200i)33-s + (1.11 − 0.641i)37-s + (−0.967 − 0.462i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.572 - 0.819i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.572 - 0.819i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2100\)    =    \(2^{2} \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $0.572 - 0.819i$
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: \((2,\ 2100,\ (\ :1/2),\ 0.572 - 0.819i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.664994027\)
\(L(\frac12)\) \(\approx\) \(2.664994027\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.56 - 0.747i)T \)
5 \( 1 \)
7 \( 1 + (-2.60 + 0.456i)T \)
good11 \( 1 + (0.698 + 0.403i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + 3.86T + 13T^{2} \)
17 \( 1 + (-1.83 - 1.05i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.70 - 1.56i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.43 - 4.21i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 6.67iT - 29T^{2} \)
31 \( 1 + (-5.65 - 3.26i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-6.75 + 3.89i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 - 8.44T + 41T^{2} \)
43 \( 1 - 0.819iT - 43T^{2} \)
47 \( 1 + (-2.40 + 1.38i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (6.67 - 11.5i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-2.86 + 4.95i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.79 + 1.03i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (9.44 + 5.45i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 10.3iT - 71T^{2} \)
73 \( 1 + (-1.54 + 2.67i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (6.76 + 11.7i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 12.8iT - 83T^{2} \)
89 \( 1 + (1.60 + 2.78i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 1.01T + 97T^{2} \)
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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

−9.165104781087954378583471697951, −8.463201066544365339490639115330, −7.64571185995516273844747246079, −7.33238584013819722200253474033, −5.96083928576449757023084750139, −4.93607306439054695561194026110, −4.44798817299207480414040973785, −3.35927193215481898547031233419, −2.44750691955657282695007670183, −1.40264009148398532686791205871, 0.907040336964787296322240973435, 2.31051621292236183776064339986, 2.70375024032653977158103034543, 4.21547527004187878772727997414, 4.72303279590003742539770280318, 5.88935956044238383039161007316, 6.83876664422762325666560853824, 7.64564982968673285352156092723, 8.112673680050547572336370430428, 8.844010699877256934883325699241

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