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Properties

Label 2-2100-7.4-c1-0-9
Degree $2$
Conductor $2100$
Sign $0.827 - 0.561i$
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  + (0.5 − 0.866i)3-s + (2.62 − 0.358i)7-s + (−0.499 − 0.866i)9-s + (−2.12 + 3.67i)11-s + 5.24·13-s + (−2.12 + 3.67i)17-s + (3.5 + 6.06i)19-s + (1 − 2.44i)21-s + (2.12 + 3.67i)23-s − 0.999·27-s − 10.2·29-s + (−3.74 + 6.48i)31-s + (2.12 + 3.67i)33-s + (−2.62 − 4.54i)37-s + (2.62 − 4.54i)39-s + ⋯
L(s)  = 1  + (0.288 − 0.499i)3-s + (0.990 − 0.135i)7-s + (−0.166 − 0.288i)9-s + (−0.639 + 1.10i)11-s + 1.45·13-s + (−0.514 + 0.891i)17-s + (0.802 + 1.39i)19-s + (0.218 − 0.534i)21-s + (0.442 + 0.766i)23-s − 0.192·27-s − 1.90·29-s + (−0.672 + 1.16i)31-s + (0.369 + 0.639i)33-s + (−0.430 − 0.746i)37-s + (0.419 − 0.727i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.827 - 0.561i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.827 - 0.561i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2100\)    =    \(2^{2} \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $0.827 - 0.561i$
Analytic conductor: \(16.7685\)
Root analytic conductor: \(4.09494\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2100} (1201, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2100,\ (\ :1/2),\ 0.827 - 0.561i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.100931739\)
\(L(\frac12)\) \(\approx\) \(2.100931739\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 + 0.866i)T \)
5 \( 1 \)
7 \( 1 + (-2.62 + 0.358i)T \)
good11 \( 1 + (2.12 - 3.67i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 5.24T + 13T^{2} \)
17 \( 1 + (2.12 - 3.67i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.5 - 6.06i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2.12 - 3.67i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 10.2T + 29T^{2} \)
31 \( 1 + (3.74 - 6.48i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (2.62 + 4.54i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 4.24T + 41T^{2} \)
43 \( 1 - 5.24T + 43T^{2} \)
47 \( 1 + (3 + 5.19i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (4.24 - 7.34i)T + (-26.5 - 45.8i)T^{2} \)
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 \( 1 + (-1.62 + 2.80i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 12.7T + 71T^{2} \)
73 \( 1 + (-0.378 + 0.655i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (5.5 + 9.52i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 1.75T + 83T^{2} \)
89 \( 1 + (-0.878 - 1.52i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 16.4T + 97T^{2} \)
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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

−8.999909430484584978690607145009, −8.336379909959379784879134695311, −7.57619085925291772740193718234, −7.14745947858092018350043635030, −5.87686851625565388110746731526, −5.37140528691835343056183353540, −4.14392719450376949873145651369, −3.46669501774726335303967577693, −1.94765346489025224868943132471, −1.45119253660053856845189412402, 0.75185262443211331860879184175, 2.24220396087466215565580559297, 3.18895994524777581416290103074, 4.10568557312784688838857269137, 5.10392193805432686547553937486, 5.60899561524717707043724983634, 6.69666953366633785458479232606, 7.71009533373659748684338099857, 8.314086770284739976603313842225, 9.039086904331800701056818020793

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