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Properties

Label 2-2100-105.89-c1-0-17
Degree $2$
Conductor $2100$
Sign $0.849 - 0.527i$
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  + (−0.348 + 1.69i)3-s + (−2.41 + 1.08i)7-s + (−2.75 − 1.18i)9-s + (1.17 − 0.675i)11-s + 4.94·13-s + (4.97 − 2.87i)17-s + (−2.84 − 1.64i)19-s + (−0.993 − 4.47i)21-s + (2.50 − 4.33i)23-s + (2.96 − 4.26i)27-s − 5.68i·29-s + (−2.45 + 1.41i)31-s + (0.738 + 2.22i)33-s + (3.33 + 1.92i)37-s + (−1.72 + 8.38i)39-s + ⋯
L(s)  = 1  + (−0.201 + 0.979i)3-s + (−0.912 + 0.409i)7-s + (−0.918 − 0.394i)9-s + (0.353 − 0.203i)11-s + 1.37·13-s + (1.20 − 0.696i)17-s + (−0.653 − 0.377i)19-s + (−0.216 − 0.976i)21-s + (0.521 − 0.903i)23-s + (0.571 − 0.820i)27-s − 1.05i·29-s + (−0.440 + 0.254i)31-s + (0.128 + 0.386i)33-s + (0.548 + 0.316i)37-s + (−0.276 + 1.34i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.849 - 0.527i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.849 - 0.527i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2100\)    =    \(2^{2} \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $0.849 - 0.527i$
Analytic conductor: \(16.7685\)
Root analytic conductor: \(4.09494\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2100} (1349, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2100,\ (\ :1/2),\ 0.849 - 0.527i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.496731095\)
\(L(\frac12)\) \(\approx\) \(1.496731095\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.348 - 1.69i)T \)
5 \( 1 \)
7 \( 1 + (2.41 - 1.08i)T \)
good11 \( 1 + (-1.17 + 0.675i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 - 4.94T + 13T^{2} \)
17 \( 1 + (-4.97 + 2.87i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.84 + 1.64i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2.50 + 4.33i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 5.68iT - 29T^{2} \)
31 \( 1 + (2.45 - 1.41i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-3.33 - 1.92i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + 3.73T + 41T^{2} \)
43 \( 1 - 4.06iT - 43T^{2} \)
47 \( 1 + (-4.92 - 2.84i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-0.730 - 1.26i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-4.34 - 7.52i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.65 - 0.954i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (4.36 - 2.51i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 3.38iT - 71T^{2} \)
73 \( 1 + (-8.16 - 14.1i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-2.41 + 4.18i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 16.6iT - 83T^{2} \)
89 \( 1 + (-8.08 + 13.9i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 12.0T + 97T^{2} \)
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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

−8.938396632349130324585434516061, −8.897974976907249849325978633419, −7.72169847875670006904319550965, −6.46321461398321035111721061075, −6.08252568999965871334892095303, −5.20419054207558355856957458314, −4.18819087258379441219191710119, −3.42656058304884093020072483557, −2.64885251113984539752614655439, −0.76448880228620222616463101642, 0.907225297233995769627243315526, 1.85985789292375584659236660165, 3.31692401953404665426275427844, 3.80483568595590630057695857978, 5.34337405142232126484934401125, 6.03181524939157064031002641967, 6.66961471326077838190362145070, 7.38253270415988790547181316064, 8.200379474372860602212526094447, 8.907482816965748721217542494999

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