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Properties

Label 2-2100-35.13-c1-0-9
Degree $2$
Conductor $2100$
Sign $0.966 - 0.257i$
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 + (−2.63 − 0.189i)7-s − 1.00i·9-s − 3·11-s + (−1.41 + 1.41i)13-s + (4.24 + 4.24i)17-s + 3.46·19-s + (−2 + 1.73i)21-s + (3.67 + 3.67i)23-s + (−0.707 − 0.707i)27-s − 9i·29-s + 6.92i·31-s + (−2.12 + 2.12i)33-s + (3.67 − 3.67i)37-s + 2.00i·39-s + ⋯
L(s)  = 1  + (0.408 − 0.408i)3-s + (−0.997 − 0.0716i)7-s − 0.333i·9-s − 0.904·11-s + (−0.392 + 0.392i)13-s + (1.02 + 1.02i)17-s + 0.794·19-s + (−0.436 + 0.377i)21-s + (0.766 + 0.766i)23-s + (−0.136 − 0.136i)27-s − 1.67i·29-s + 1.24i·31-s + (−0.369 + 0.369i)33-s + (0.604 − 0.604i)37-s + 0.320i·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.966 - 0.257i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.966 - 0.257i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2100\)    =    \(2^{2} \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $0.966 - 0.257i$
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: \((2,\ 2100,\ (\ :1/2),\ 0.966 - 0.257i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.636569121\)
\(L(\frac12)\) \(\approx\) \(1.636569121\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.63 + 0.189i)T \)
good11 \( 1 + 3T + 11T^{2} \)
13 \( 1 + (1.41 - 1.41i)T - 13iT^{2} \)
17 \( 1 + (-4.24 - 4.24i)T + 17iT^{2} \)
19 \( 1 - 3.46T + 19T^{2} \)
23 \( 1 + (-3.67 - 3.67i)T + 23iT^{2} \)
29 \( 1 + 9iT - 29T^{2} \)
31 \( 1 - 6.92iT - 31T^{2} \)
37 \( 1 + (-3.67 + 3.67i)T - 37iT^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 + (-8.57 - 8.57i)T + 43iT^{2} \)
47 \( 1 + (-4.24 - 4.24i)T + 47iT^{2} \)
53 \( 1 + 53iT^{2} \)
59 \( 1 - 10.3T + 59T^{2} \)
61 \( 1 + 6.92iT - 61T^{2} \)
67 \( 1 + (-8.57 + 8.57i)T - 67iT^{2} \)
71 \( 1 - 3T + 71T^{2} \)
73 \( 1 + (-1.41 + 1.41i)T - 73iT^{2} \)
79 \( 1 + iT - 79T^{2} \)
83 \( 1 + (8.48 - 8.48i)T - 83iT^{2} \)
89 \( 1 + 10.3T + 89T^{2} \)
97 \( 1 + (-2.82 - 2.82i)T + 97iT^{2} \)
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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

−9.355634654506712616039882284433, −8.158116970902628391885939476958, −7.66917763436091607754684634900, −6.88749266923660522325332734927, −6.00854872646675687260049608347, −5.30434772959619992037723468222, −4.04884231994450718252856823438, −3.19087276596127390759346006235, −2.40430976968436423410155049052, −0.991929996041089219402562143838, 0.68662947406279345236234715939, 2.61536066038738878127445891712, 3.01608045353504094043530718889, 4.05945129825690415327311256718, 5.25670979160756385577609525921, 5.60483313288257038851561737626, 7.00423997498214721961892466095, 7.40306638612315788993879984760, 8.390398712241730773569242342817, 9.163784699172511157988322428410

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