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Properties

Label 2-2100-21.17-c1-0-35
Degree $2$
Conductor $2100$
Sign $0.925 + 0.379i$
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  + (1.63 − 0.576i)3-s + (1.73 + 1.99i)7-s + (2.33 − 1.88i)9-s + (3.38 + 1.95i)11-s − 6.06i·13-s + (1.53 − 2.65i)17-s + (−2.94 + 1.70i)19-s + (3.98 + 2.26i)21-s + (2.48 − 1.43i)23-s + (2.72 − 4.42i)27-s + 7.97i·29-s + (−5.63 − 3.25i)31-s + (6.64 + 1.23i)33-s + (0.0654 + 0.113i)37-s + (−3.49 − 9.90i)39-s + ⋯
L(s)  = 1  + (0.942 − 0.333i)3-s + (0.655 + 0.755i)7-s + (0.778 − 0.628i)9-s + (1.01 + 0.588i)11-s − 1.68i·13-s + (0.371 − 0.643i)17-s + (−0.676 + 0.390i)19-s + (0.869 + 0.493i)21-s + (0.518 − 0.299i)23-s + (0.524 − 0.851i)27-s + 1.48i·29-s + (−1.01 − 0.583i)31-s + (1.15 + 0.215i)33-s + (0.0107 + 0.0186i)37-s + (−0.560 − 1.58i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.925 + 0.379i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.925 + 0.379i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2100\)    =    \(2^{2} \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $0.925 + 0.379i$
Analytic conductor: \(16.7685\)
Root analytic conductor: \(4.09494\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2100} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2100,\ (\ :1/2),\ 0.925 + 0.379i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.964174042\)
\(L(\frac12)\) \(\approx\) \(2.964174042\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.63 + 0.576i)T \)
5 \( 1 \)
7 \( 1 + (-1.73 - 1.99i)T \)
good11 \( 1 + (-3.38 - 1.95i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + 6.06iT - 13T^{2} \)
17 \( 1 + (-1.53 + 2.65i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.94 - 1.70i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.48 + 1.43i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 7.97iT - 29T^{2} \)
31 \( 1 + (5.63 + 3.25i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.0654 - 0.113i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 12.3T + 41T^{2} \)
43 \( 1 + 4.43T + 43T^{2} \)
47 \( 1 + (5.02 + 8.71i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-4.64 - 2.67i)T + (26.5 + 45.8i)T^{2} \)
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 \( 1 + (7.99 - 13.8i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 3.63iT - 71T^{2} \)
73 \( 1 + (-6.72 - 3.88i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (1.22 + 2.11i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 7.63T + 83T^{2} \)
89 \( 1 + (4.11 + 7.13i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 5.74iT - 97T^{2} \)
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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

−8.895387390165669223415610869191, −8.382853824860348638989531199094, −7.56650662934021860233817678332, −6.95366899513787398432206361289, −5.86618732886327861652988556369, −5.06312209220831100912861587009, −4.00374134820333694406871511561, −3.07986244032309568858054137574, −2.19099450778344596719841289235, −1.12958112266404159432763636169, 1.31920536836106230953724010478, 2.18288078549985607168031452429, 3.57709238627251487827551834044, 4.12908130841252346591061621628, 4.78775857892317263318187907447, 6.17066445629524218114683397067, 6.91987540415341502483608149318, 7.71372015568626784724392925360, 8.445912292959141649819392586574, 9.194260791608403638691309001055

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