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Properties

Label 2-2100-15.2-c1-0-14
Degree $2$
Conductor $2100$
Sign $0.824 - 0.565i$
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  + (−1.61 + 0.625i)3-s + (−0.707 − 0.707i)7-s + (2.21 − 2.01i)9-s + 1.94i·11-s + (−0.405 + 0.405i)13-s + (0.0645 − 0.0645i)17-s − 0.513i·19-s + (1.58 + 0.700i)21-s + (−2.07 − 2.07i)23-s + (−2.32 + 4.64i)27-s − 1.78·29-s + 4.83·31-s + (−1.21 − 3.14i)33-s + (5.59 + 5.59i)37-s + (0.401 − 0.908i)39-s + ⋯
L(s)  = 1  + (−0.932 + 0.360i)3-s + (−0.267 − 0.267i)7-s + (0.739 − 0.673i)9-s + 0.586i·11-s + (−0.112 + 0.112i)13-s + (0.0156 − 0.0156i)17-s − 0.117i·19-s + (0.345 + 0.152i)21-s + (−0.432 − 0.432i)23-s + (−0.446 + 0.894i)27-s − 0.331·29-s + 0.868·31-s + (−0.211 − 0.546i)33-s + (0.919 + 0.919i)37-s + (0.0642 − 0.145i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.824 - 0.565i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.824 - 0.565i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2100\)    =    \(2^{2} \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $0.824 - 0.565i$
Analytic conductor: \(16.7685\)
Root analytic conductor: \(4.09494\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2100} (1457, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2100,\ (\ :1/2),\ 0.824 - 0.565i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.083180605\)
\(L(\frac12)\) \(\approx\) \(1.083180605\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.61 - 0.625i)T \)
5 \( 1 \)
7 \( 1 + (0.707 + 0.707i)T \)
good11 \( 1 - 1.94iT - 11T^{2} \)
13 \( 1 + (0.405 - 0.405i)T - 13iT^{2} \)
17 \( 1 + (-0.0645 + 0.0645i)T - 17iT^{2} \)
19 \( 1 + 0.513iT - 19T^{2} \)
23 \( 1 + (2.07 + 2.07i)T + 23iT^{2} \)
29 \( 1 + 1.78T + 29T^{2} \)
31 \( 1 - 4.83T + 31T^{2} \)
37 \( 1 + (-5.59 - 5.59i)T + 37iT^{2} \)
41 \( 1 + 12.4iT - 41T^{2} \)
43 \( 1 + (4.95 - 4.95i)T - 43iT^{2} \)
47 \( 1 + (-6.03 + 6.03i)T - 47iT^{2} \)
53 \( 1 + (-2.74 - 2.74i)T + 53iT^{2} \)
59 \( 1 + 12.3T + 59T^{2} \)
61 \( 1 - 12.0T + 61T^{2} \)
67 \( 1 + (-0.645 - 0.645i)T + 67iT^{2} \)
71 \( 1 - 10.2iT - 71T^{2} \)
73 \( 1 + (-6.71 + 6.71i)T - 73iT^{2} \)
79 \( 1 - 9.75iT - 79T^{2} \)
83 \( 1 + (-11.7 - 11.7i)T + 83iT^{2} \)
89 \( 1 - 2.42T + 89T^{2} \)
97 \( 1 + (-12.0 - 12.0i)T + 97iT^{2} \)
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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

−9.420827693784750818425780126398, −8.443431918828459055121906022459, −7.42988892572960194222111945052, −6.74885680199308622872482890069, −6.05086516643359037473037279620, −5.13489423803865211973296141265, −4.40061130719897720835554947782, −3.60176547366118891175094021528, −2.24109477126954064124367912077, −0.804432509954179981605948516755, 0.64209180181296663800914034721, 1.94082811321060267384741637055, 3.14181482884873849475822798810, 4.27731744795797576101385758809, 5.16490222444726353566671322158, 6.00338310304364169457922185928, 6.43858192518050998285051916882, 7.49950740142525439591882308449, 8.062787342422325704272575239954, 9.096679548615301878216613144799

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