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Properties

Label 2-2100-15.8-c1-0-26
Degree $2$
Conductor $2100$
Sign $0.0748 + 0.997i$
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  + (0.522 + 1.65i)3-s + (0.707 − 0.707i)7-s + (−2.45 + 1.72i)9-s + 3.21i·11-s + (−3.87 − 3.87i)13-s + (−4.34 − 4.34i)17-s − 6.27i·19-s + (1.53 + 0.797i)21-s + (2.64 − 2.64i)23-s + (−4.13 − 3.14i)27-s − 8.25·29-s + 2.30·31-s + (−5.30 + 1.67i)33-s + (5.79 − 5.79i)37-s + (4.36 − 8.41i)39-s + ⋯
L(s)  = 1  + (0.301 + 0.953i)3-s + (0.267 − 0.267i)7-s + (−0.817 + 0.575i)9-s + 0.968i·11-s + (−1.07 − 1.07i)13-s + (−1.05 − 1.05i)17-s − 1.44i·19-s + (0.335 + 0.174i)21-s + (0.550 − 0.550i)23-s + (−0.795 − 0.605i)27-s − 1.53·29-s + 0.414·31-s + (−0.922 + 0.292i)33-s + (0.952 − 0.952i)37-s + (0.699 − 1.34i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0748 + 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0748 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2100\)    =    \(2^{2} \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $0.0748 + 0.997i$
Analytic conductor: \(16.7685\)
Root analytic conductor: \(4.09494\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2100} (1793, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2100,\ (\ :1/2),\ 0.0748 + 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8160498055\)
\(L(\frac12)\) \(\approx\) \(0.8160498055\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.522 - 1.65i)T \)
5 \( 1 \)
7 \( 1 + (-0.707 + 0.707i)T \)
good11 \( 1 - 3.21iT - 11T^{2} \)
13 \( 1 + (3.87 + 3.87i)T + 13iT^{2} \)
17 \( 1 + (4.34 + 4.34i)T + 17iT^{2} \)
19 \( 1 + 6.27iT - 19T^{2} \)
23 \( 1 + (-2.64 + 2.64i)T - 23iT^{2} \)
29 \( 1 + 8.25T + 29T^{2} \)
31 \( 1 - 2.30T + 31T^{2} \)
37 \( 1 + (-5.79 + 5.79i)T - 37iT^{2} \)
41 \( 1 - 9.59iT - 41T^{2} \)
43 \( 1 + (3.88 + 3.88i)T + 43iT^{2} \)
47 \( 1 + (1.57 + 1.57i)T + 47iT^{2} \)
53 \( 1 + (-2.48 + 2.48i)T - 53iT^{2} \)
59 \( 1 + 0.317T + 59T^{2} \)
61 \( 1 + 3.94T + 61T^{2} \)
67 \( 1 + (-4.21 + 4.21i)T - 67iT^{2} \)
71 \( 1 + 9.52iT - 71T^{2} \)
73 \( 1 + (-1.46 - 1.46i)T + 73iT^{2} \)
79 \( 1 - 12.9iT - 79T^{2} \)
83 \( 1 + (9.41 - 9.41i)T - 83iT^{2} \)
89 \( 1 + 4.62T + 89T^{2} \)
97 \( 1 + (5.42 - 5.42i)T - 97iT^{2} \)
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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

−9.216604390442902018465423152858, −8.171398443510056330891837632279, −7.41332929533795085426891333828, −6.74737457816495375534729588480, −5.33648481147694215355136117820, −4.84923499235767587379310064723, −4.20317553363105237266212349941, −2.90424733256511112653996123317, −2.30936154727018190317609050370, −0.25832933335603232619174589690, 1.49410706015112272224343861786, 2.25193398718755129160136124081, 3.38396813832893838449664569859, 4.33497139949892133626534427941, 5.57690580796678635805282854919, 6.17393593682786017633409177513, 7.00902967986350717915730331301, 7.75808842773441870232558136356, 8.482088644243590086348456482974, 9.062879061293704300178911308051

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