Properties

Label 2-2100-35.17-c1-0-13
Degree $2$
Conductor $2100$
Sign $0.978 + 0.204i$
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  + (−0.965 + 0.258i)3-s + (−0.106 + 2.64i)7-s + (0.866 − 0.499i)9-s + (3.02 − 5.23i)11-s + (2.18 + 2.18i)13-s + (−1.66 − 6.21i)17-s + (3.37 + 5.84i)19-s + (−0.580 − 2.58i)21-s + (−3.78 − 1.01i)23-s + (−0.707 + 0.707i)27-s − 2.45i·29-s + (−2.81 − 1.62i)31-s + (−1.56 + 5.83i)33-s + (−0.159 + 0.594i)37-s + (−2.68 − 1.54i)39-s + ⋯
L(s)  = 1  + (−0.557 + 0.149i)3-s + (−0.0404 + 0.999i)7-s + (0.288 − 0.166i)9-s + (0.910 − 1.57i)11-s + (0.607 + 0.607i)13-s + (−0.404 − 1.50i)17-s + (0.773 + 1.34i)19-s + (−0.126 − 0.563i)21-s + (−0.790 − 0.211i)23-s + (−0.136 + 0.136i)27-s − 0.456i·29-s + (−0.505 − 0.292i)31-s + (−0.272 + 1.01i)33-s + (−0.0261 + 0.0977i)37-s + (−0.429 − 0.247i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.978 + 0.204i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.978 + 0.204i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2100\)    =    \(2^{2} \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $0.978 + 0.204i$
Analytic conductor: \(16.7685\)
Root analytic conductor: \(4.09494\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2100} (157, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2100,\ (\ :1/2),\ 0.978 + 0.204i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.499010724\)
\(L(\frac12)\) \(\approx\) \(1.499010724\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.965 - 0.258i)T \)
5 \( 1 \)
7 \( 1 + (0.106 - 2.64i)T \)
good11 \( 1 + (-3.02 + 5.23i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-2.18 - 2.18i)T + 13iT^{2} \)
17 \( 1 + (1.66 + 6.21i)T + (-14.7 + 8.5i)T^{2} \)
19 \( 1 + (-3.37 - 5.84i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (3.78 + 1.01i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 + 2.45iT - 29T^{2} \)
31 \( 1 + (2.81 + 1.62i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.159 - 0.594i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + 6.77iT - 41T^{2} \)
43 \( 1 + (-5.33 + 5.33i)T - 43iT^{2} \)
47 \( 1 + (-6.44 - 1.72i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (-0.905 - 3.38i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (2.28 - 3.95i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (3.38 - 1.95i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-11.0 + 2.96i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 - 5.40T + 71T^{2} \)
73 \( 1 + (-6.99 + 1.87i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (-8.00 + 4.61i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (0.451 + 0.451i)T + 83iT^{2} \)
89 \( 1 + (-6.04 - 10.4i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-9.62 + 9.62i)T - 97iT^{2} \)
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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

−9.134113776950603071547871695008, −8.481546109743601973564660697446, −7.51404347977134786939018649936, −6.46011639910457457526987815471, −5.89541168181678594845978065314, −5.34184175710218479614552575501, −4.09752719962044105672697402658, −3.37063359394477046757550967647, −2.12158747134123227559070947677, −0.75044956609017258742284837005, 0.987213471155069231810663863772, 1.96892179771066803085345420534, 3.54702054428576514154502689861, 4.26775207931255082492719861778, 5.02603752127602293703428194284, 6.15551639809871982130961310265, 6.78705451310283346034688891052, 7.42506828969735545880332682549, 8.220524512823673720089662843491, 9.315483087243202852427930454730

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