Properties

Label 2-2100-35.13-c1-0-8
Degree $2$
Conductor $2100$
Sign $0.669 - 0.742i$
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  + (0.707 − 0.707i)3-s + (1.71 + 2.01i)7-s − 1.00i·9-s − 4.27·11-s + (−2.31 + 2.31i)13-s + (−1.41 − 1.41i)17-s + 6.50·19-s + (2.63 + 0.209i)21-s + (3.37 + 3.37i)23-s + (−0.707 − 0.707i)27-s + 8.27i·29-s − 3.04i·31-s + (−3.02 + 3.02i)33-s + (7.68 − 7.68i)37-s + 3.27i·39-s + ⋯
L(s)  = 1  + (0.408 − 0.408i)3-s + (0.648 + 0.760i)7-s − 0.333i·9-s − 1.28·11-s + (−0.642 + 0.642i)13-s + (−0.342 − 0.342i)17-s + 1.49·19-s + (0.575 + 0.0456i)21-s + (0.704 + 0.704i)23-s + (−0.136 − 0.136i)27-s + 1.53i·29-s − 0.546i·31-s + (−0.526 + 0.526i)33-s + (1.26 − 1.26i)37-s + 0.524i·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.669 - 0.742i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.669 - 0.742i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2100\)    =    \(2^{2} \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $0.669 - 0.742i$
Analytic conductor: \(16.7685\)
Root analytic conductor: \(4.09494\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2100} (1693, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2100,\ (\ :1/2),\ 0.669 - 0.742i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.857853358\)
\(L(\frac12)\) \(\approx\) \(1.857853358\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.707 + 0.707i)T \)
5 \( 1 \)
7 \( 1 + (-1.71 - 2.01i)T \)
good11 \( 1 + 4.27T + 11T^{2} \)
13 \( 1 + (2.31 - 2.31i)T - 13iT^{2} \)
17 \( 1 + (1.41 + 1.41i)T + 17iT^{2} \)
19 \( 1 - 6.50T + 19T^{2} \)
23 \( 1 + (-3.37 - 3.37i)T + 23iT^{2} \)
29 \( 1 - 8.27iT - 29T^{2} \)
31 \( 1 + 3.04iT - 31T^{2} \)
37 \( 1 + (-7.68 + 7.68i)T - 37iT^{2} \)
41 \( 1 - 12.1iT - 41T^{2} \)
43 \( 1 + (-5.53 - 5.53i)T + 43iT^{2} \)
47 \( 1 + (1.41 + 1.41i)T + 47iT^{2} \)
53 \( 1 + (-8.61 - 8.61i)T + 53iT^{2} \)
59 \( 1 + 8.71T + 59T^{2} \)
61 \( 1 - 3.04iT - 61T^{2} \)
67 \( 1 + (3.08 - 3.08i)T - 67iT^{2} \)
71 \( 1 - 1.72T + 71T^{2} \)
73 \( 1 + (-7.07 + 7.07i)T - 73iT^{2} \)
79 \( 1 - 10.8iT - 79T^{2} \)
83 \( 1 + (-1.02 + 1.02i)T - 83iT^{2} \)
89 \( 1 + 7.88T + 89T^{2} \)
97 \( 1 + (1.92 + 1.92i)T + 97iT^{2} \)
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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

−9.322257895084440052492412135335, −8.352734194518134732800106060953, −7.56703933445834379156864386628, −7.22247738803382568636894193352, −5.93113659330687508421567033178, −5.22697864164434599913446756123, −4.50457039359947996814408056163, −3.02551938720205145247017950829, −2.48454410660864679128077540556, −1.28141827937439763245567251564, 0.66104830716655646479321570534, 2.24930681751509527960628746859, 3.07756275440069210848893837813, 4.14792840870599123459828085364, 4.98225364829551361500081062874, 5.53599759659198348952164718664, 6.86000694042695324315425046774, 7.74273256573321051279285355541, 7.989837642104455948583234501269, 8.987957145802736362574337114110

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