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Properties

Label 2-2100-35.33-c1-0-12
Degree $2$
Conductor $2100$
Sign $0.985 - 0.171i$
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  + (0.965 + 0.258i)3-s + (−2.48 − 0.920i)7-s + (0.866 + 0.499i)9-s + (1.65 + 2.87i)11-s + (1.05 − 1.05i)13-s + (0.858 − 3.20i)17-s + (−1.10 + 1.91i)19-s + (−2.15 − 1.53i)21-s + (6.83 − 1.83i)23-s + (0.707 + 0.707i)27-s + (1.56 − 0.902i)31-s + (0.858 + 3.20i)33-s + (−0.448 − 1.67i)37-s + (1.29 − 0.747i)39-s + 1.33i·41-s + ⋯
L(s)  = 1  + (0.557 + 0.149i)3-s + (−0.937 − 0.348i)7-s + (0.288 + 0.166i)9-s + (0.499 + 0.865i)11-s + (0.293 − 0.293i)13-s + (0.208 − 0.776i)17-s + (−0.253 + 0.438i)19-s + (−0.470 − 0.334i)21-s + (1.42 − 0.381i)23-s + (0.136 + 0.136i)27-s + (0.280 − 0.162i)31-s + (0.149 + 0.557i)33-s + (−0.0736 − 0.275i)37-s + (0.207 − 0.119i)39-s + 0.207i·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.985 - 0.171i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.985 - 0.171i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2100\)    =    \(2^{2} \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $0.985 - 0.171i$
Analytic conductor: \(16.7685\)
Root analytic conductor: \(4.09494\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2100} (1993, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2100,\ (\ :1/2),\ 0.985 - 0.171i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.070131771\)
\(L(\frac12)\) \(\approx\) \(2.070131771\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.965 - 0.258i)T \)
5 \( 1 \)
7 \( 1 + (2.48 + 0.920i)T \)
good11 \( 1 + (-1.65 - 2.87i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.05 + 1.05i)T - 13iT^{2} \)
17 \( 1 + (-0.858 + 3.20i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (1.10 - 1.91i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-6.83 + 1.83i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 - 29T^{2} \)
31 \( 1 + (-1.56 + 0.902i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.448 + 1.67i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 - 1.33iT - 41T^{2} \)
43 \( 1 + (-6.84 - 6.84i)T + 43iT^{2} \)
47 \( 1 + (-4.17 + 1.11i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (0.797 - 2.97i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (-0.665 - 1.15i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (4.62 + 2.67i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.13 - 0.570i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 - 4.98T + 71T^{2} \)
73 \( 1 + (-10.2 - 2.74i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (-13.8 - 7.96i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-10.1 + 10.1i)T - 83iT^{2} \)
89 \( 1 + (2.87 - 4.97i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-3.53 - 3.53i)T + 97iT^{2} \)
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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

−9.437112961419975197659992504650, −8.405048064283949107891005120131, −7.49202024196394422038419505063, −6.89874000678670129171621343890, −6.11276868023169696002841462186, −4.97570005447523347826067607531, −4.12893859871725787884975672711, −3.27803835204466185529144395418, −2.42372225786628006746013108881, −0.974654929750639101468230328414, 0.926787315508710172302816806768, 2.30478104889015965193625737502, 3.30313388289829481415770777750, 3.86459125661726593774670501827, 5.13519944752143810198738777598, 6.12657022288703940235146779369, 6.65732320472963210223711816440, 7.52640204702507354914058707064, 8.555919200019041561193782602302, 8.984188365578353970160979820710

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