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Properties

Label 2-2100-35.33-c1-0-3
Degree $2$
Conductor $2100$
Sign $-0.365 - 0.930i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.365 - 0.930i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.365 - 0.930i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2100\)    =    \(2^{2} \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $-0.365 - 0.930i$
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: \((2,\ 2100,\ (\ :1/2),\ -0.365 - 0.930i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.088267861\)
\(L(\frac12)\) \(\approx\) \(1.088267861\)
\(L(\frac{3}{2})\) 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

−9.437656578975121191147938239373, −8.393608293466244742837639120929, −7.87798709023115038261162142712, −6.93175842837515035183792500679, −6.20541892582690956789089987615, −5.36205395338456408980132867868, −4.53772720115137182083415455095, −3.82805565749001174033944296583, −2.17954905147482270443706860337, −1.52934097496452032359283217736, 0.41173012331425965062997938933, 1.68241983824388698753248976229, 2.98806168973620434840635446166, 4.15474486533003207729703417005, 4.78449454288586930550709087466, 5.67499334697745641221591178048, 6.44229810209385440979917935061, 7.30079199365857672780601261207, 8.120996203232210480125908270342, 8.779470955795890634805858524787

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