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Properties

Label 2-2100-35.13-c1-0-15
Degree $2$
Conductor $2100$
Sign $0.178 + 0.983i$
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 + (−1.05 − 2.42i)7-s − 1.00i·9-s + 0.301·11-s + (−4.02 + 4.02i)13-s + (0.233 + 0.233i)17-s + 3.05·19-s + (2.46 + 0.966i)21-s + (4.22 + 4.22i)23-s + (0.707 + 0.707i)27-s − 2.19i·29-s − 0.852i·31-s + (−0.213 + 0.213i)33-s + (6.87 − 6.87i)37-s − 5.68i·39-s + ⋯
L(s)  = 1  + (−0.408 + 0.408i)3-s + (−0.399 − 0.916i)7-s − 0.333i·9-s + 0.0910·11-s + (−1.11 + 1.11i)13-s + (0.0567 + 0.0567i)17-s + 0.701·19-s + (0.537 + 0.210i)21-s + (0.881 + 0.881i)23-s + (0.136 + 0.136i)27-s − 0.406i·29-s − 0.153i·31-s + (−0.0371 + 0.0371i)33-s + (1.13 − 1.13i)37-s − 0.910i·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.178 + 0.983i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.178 + 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2100\)    =    \(2^{2} \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $0.178 + 0.983i$
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: \((2,\ 2100,\ (\ :1/2),\ 0.178 + 0.983i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9178393595\)
\(L(\frac12)\) \(\approx\) \(0.9178393595\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.05 + 2.42i)T \)
good11 \( 1 - 0.301T + 11T^{2} \)
13 \( 1 + (4.02 - 4.02i)T - 13iT^{2} \)
17 \( 1 + (-0.233 - 0.233i)T + 17iT^{2} \)
19 \( 1 - 3.05T + 19T^{2} \)
23 \( 1 + (-4.22 - 4.22i)T + 23iT^{2} \)
29 \( 1 + 2.19iT - 29T^{2} \)
31 \( 1 + 0.852iT - 31T^{2} \)
37 \( 1 + (-6.87 + 6.87i)T - 37iT^{2} \)
41 \( 1 + 6.83iT - 41T^{2} \)
43 \( 1 + (8.32 + 8.32i)T + 43iT^{2} \)
47 \( 1 + (6.40 + 6.40i)T + 47iT^{2} \)
53 \( 1 + (5.45 + 5.45i)T + 53iT^{2} \)
59 \( 1 + 1.30T + 59T^{2} \)
61 \( 1 - 3.72iT - 61T^{2} \)
67 \( 1 + (-4.16 + 4.16i)T - 67iT^{2} \)
71 \( 1 - 9.13T + 71T^{2} \)
73 \( 1 + (-2.98 + 2.98i)T - 73iT^{2} \)
79 \( 1 - 0.382iT - 79T^{2} \)
83 \( 1 + (-11.0 + 11.0i)T - 83iT^{2} \)
89 \( 1 + 17.2T + 89T^{2} \)
97 \( 1 + (3.75 + 3.75i)T + 97iT^{2} \)
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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

−9.332667248826376945965809990164, −8.079095529711345020542690586478, −7.10249155901663966799204069702, −6.84176955074369727827847874470, −5.64830750098095488645607889034, −4.87358907823612366283453311779, −4.04992724774775284571367341327, −3.25612144530368486980380082443, −1.87596916181905409867739486499, −0.38146701130409877253408877316, 1.13128117782194681724341451180, 2.62306364750156074997584702607, 3.12535830174615263069398360896, 4.79571533602187303689406150709, 5.20801217500219706097979702933, 6.22993697198966214138093613834, 6.77371002930021892318627368254, 7.84732832114217945302583271786, 8.287648886775555180395235171356, 9.508286032472720695988841106655

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