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Properties

Label 2-2100-105.59-c1-0-36
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: $\chi_{2100} (1949, \cdot )$
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

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

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