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Properties

Label 2-2100-35.9-c1-0-6
Degree $2$
Conductor $2100$
Sign $0.872 - 0.488i$
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  + (−0.866 + 0.5i)3-s + (0.358 − 2.62i)7-s + (0.499 − 0.866i)9-s + (−2.12 − 3.67i)11-s + 5.24i·13-s + (−3.67 + 2.12i)17-s + (−3.5 + 6.06i)19-s + (1 + 2.44i)21-s + (3.67 + 2.12i)23-s + 0.999i·27-s + 10.2·29-s + (−3.74 − 6.48i)31-s + (3.67 + 2.12i)33-s + (4.54 + 2.62i)37-s + (−2.62 − 4.54i)39-s + ⋯
L(s)  = 1  + (−0.499 + 0.288i)3-s + (0.135 − 0.990i)7-s + (0.166 − 0.288i)9-s + (−0.639 − 1.10i)11-s + 1.45i·13-s + (−0.891 + 0.514i)17-s + (−0.802 + 1.39i)19-s + (0.218 + 0.534i)21-s + (0.766 + 0.442i)23-s + 0.192i·27-s + 1.90·29-s + (−0.672 − 1.16i)31-s + (0.639 + 0.369i)33-s + (0.746 + 0.430i)37-s + (−0.419 − 0.727i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.872 - 0.488i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.872 - 0.488i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2100\)    =    \(2^{2} \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $0.872 - 0.488i$
Analytic conductor: \(16.7685\)
Root analytic conductor: \(4.09494\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2100} (1549, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2100,\ (\ :1/2),\ 0.872 - 0.488i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.257529147\)
\(L(\frac12)\) \(\approx\) \(1.257529147\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.866 - 0.5i)T \)
5 \( 1 \)
7 \( 1 + (-0.358 + 2.62i)T \)
good11 \( 1 + (2.12 + 3.67i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 5.24iT - 13T^{2} \)
17 \( 1 + (3.67 - 2.12i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.5 - 6.06i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3.67 - 2.12i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 10.2T + 29T^{2} \)
31 \( 1 + (3.74 + 6.48i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-4.54 - 2.62i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 - 4.24T + 41T^{2} \)
43 \( 1 - 5.24iT - 43T^{2} \)
47 \( 1 + (-5.19 - 3i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-7.34 + 4.24i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (0.878 + 1.52i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-6.24 + 10.8i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.80 + 1.62i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 12.7T + 71T^{2} \)
73 \( 1 + (0.655 - 0.378i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-5.5 + 9.52i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 1.75iT - 83T^{2} \)
89 \( 1 + (0.878 - 1.52i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 16.4iT - 97T^{2} \)
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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

−9.214080451090847893068918511926, −8.341544978706464109358293768512, −7.69306759437575804995998192955, −6.52253163270124096734743314940, −6.26291567090902131164463451949, −5.06600561114763692674992990865, −4.24214658875553611125820391716, −3.64454348579112142209402078507, −2.21643398855174535431902912868, −0.881768654248735594349130122691, 0.64081314050773020178865364162, 2.34851132685536304594667467712, 2.76324959252780837221577245797, 4.49510425163848600928812495716, 5.05731459222217955819341618167, 5.76728856377984748615320957773, 6.83196179958795407171266956940, 7.27472980833739444920638863450, 8.434611062770054767808120900432, 8.833664953785612450177395436559

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