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Properties

Label 2-2100-105.89-c1-0-6
Degree $2$
Conductor $2100$
Sign $-0.692 - 0.721i$
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  + (−1.42 + 0.979i)3-s + (2.60 + 0.456i)7-s + (1.08 − 2.79i)9-s + (0.698 − 0.403i)11-s − 3.86·13-s + (−1.83 + 1.05i)17-s + (−2.70 − 1.56i)19-s + (−4.17 + 1.89i)21-s + (−2.43 + 4.21i)23-s + (1.19 + 5.05i)27-s + 6.67i·29-s + (5.65 − 3.26i)31-s + (−0.603 + 1.25i)33-s + (6.75 + 3.89i)37-s + (5.52 − 3.78i)39-s + ⋯
L(s)  = 1  + (−0.824 + 0.565i)3-s + (0.985 + 0.172i)7-s + (0.360 − 0.932i)9-s + (0.210 − 0.121i)11-s − 1.07·13-s + (−0.445 + 0.257i)17-s + (−0.620 − 0.358i)19-s + (−0.910 + 0.414i)21-s + (−0.508 + 0.879i)23-s + (0.229 + 0.973i)27-s + 1.23i·29-s + (1.01 − 0.585i)31-s + (−0.104 + 0.219i)33-s + (1.11 + 0.641i)37-s + (0.884 − 0.606i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.692 - 0.721i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.692 - 0.721i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2100\)    =    \(2^{2} \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $-0.692 - 0.721i$
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: \((2,\ 2100,\ (\ :1/2),\ -0.692 - 0.721i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8236371607\)
\(L(\frac12)\) \(\approx\) \(0.8236371607\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.42 - 0.979i)T \)
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

−9.456073239005001882729439013891, −8.709300960208630333610451176486, −7.84206589163819681899460558790, −6.96135791627677676855825123252, −6.16564506320351014189598145726, −5.25532419745707130632595867927, −4.68550176854216661603129493277, −3.91588562607865554617860837286, −2.58023752394931943440450851165, −1.30971223452550341461920123617, 0.33588541287406802294405680969, 1.72699655966733592182135903745, 2.53764814389902836855256690150, 4.31144425195392275163716501651, 4.70440212526910363037263864177, 5.67080894144932312699620637035, 6.50484421887333921042972734153, 7.20805537361697282440417824081, 7.986447583405163299288242002668, 8.554317015742225538875644789339

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