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Properties

Label 2-2100-105.89-c1-0-11
Degree $2$
Conductor $2100$
Sign $0.0739 - 0.997i$
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  + (−0.576 + 1.63i)3-s + (−1.99 − 1.73i)7-s + (−2.33 − 1.88i)9-s + (3.38 − 1.95i)11-s − 6.06·13-s + (2.65 − 1.53i)17-s + (2.94 + 1.70i)19-s + (3.98 − 2.26i)21-s + (−1.43 + 2.48i)23-s + (4.42 − 2.72i)27-s + 7.97i·29-s + (−5.63 + 3.25i)31-s + (1.23 + 6.64i)33-s + (−0.113 − 0.0654i)37-s + (3.49 − 9.90i)39-s + ⋯
L(s)  = 1  + (−0.333 + 0.942i)3-s + (−0.755 − 0.655i)7-s + (−0.778 − 0.628i)9-s + (1.01 − 0.588i)11-s − 1.68·13-s + (0.643 − 0.371i)17-s + (0.676 + 0.390i)19-s + (0.869 − 0.493i)21-s + (−0.299 + 0.518i)23-s + (0.851 − 0.524i)27-s + 1.48i·29-s + (−1.01 + 0.583i)31-s + (0.215 + 1.15i)33-s + (−0.0186 − 0.0107i)37-s + (0.560 − 1.58i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0739 - 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0739 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2100\)    =    \(2^{2} \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $0.0739 - 0.997i$
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: \((2,\ 2100,\ (\ :1/2),\ 0.0739 - 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.089086821\)
\(L(\frac12)\) \(\approx\) \(1.089086821\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.576 - 1.63i)T \)
5 \( 1 \)
7 \( 1 + (1.99 + 1.73i)T \)
good11 \( 1 + (-3.38 + 1.95i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + 6.06T + 13T^{2} \)
17 \( 1 + (-2.65 + 1.53i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.94 - 1.70i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.43 - 2.48i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 7.97iT - 29T^{2} \)
31 \( 1 + (5.63 - 3.25i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.113 + 0.0654i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 - 12.3T + 41T^{2} \)
43 \( 1 + 4.43iT - 43T^{2} \)
47 \( 1 + (-8.71 - 5.02i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-2.67 - 4.64i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-1.28 - 2.23i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-7.44 - 4.30i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (13.8 - 7.99i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 3.63iT - 71T^{2} \)
73 \( 1 + (-3.88 - 6.72i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-1.22 + 2.11i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 7.63iT - 83T^{2} \)
89 \( 1 + (-4.11 + 7.13i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 5.74T + 97T^{2} \)
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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

−9.345432630020594836693855812066, −8.911987918698013382266417094457, −7.48228891918023274090033525356, −7.08287917912834781467812245683, −5.92726334499385699682363438305, −5.36217896700291296785293818019, −4.32101967081611623564407955877, −3.59683454967204434103474921439, −2.82001978358544303825257829697, −0.960024604831954970430878687169, 0.50236510722992995396557395658, 2.03668173137054405553280830175, 2.69455857358009732760660522308, 3.99563112620725462930985675638, 5.10116622121285536339591997305, 5.89947954025725216371268343331, 6.56287493371998898563197802820, 7.38364762913606132166343907721, 7.85790279896701341371530328317, 9.082775658554600509044389136793

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