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Properties

Label 2-2100-15.8-c1-0-27
Degree $2$
Conductor $2100$
Sign $-0.546 + 0.837i$
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  + (−0.587 + 1.62i)3-s + (0.707 − 0.707i)7-s + (−2.30 − 1.91i)9-s − 2.43i·11-s + (2.98 + 2.98i)13-s + (−4.94 − 4.94i)17-s + 1.61i·19-s + (0.736 + 1.56i)21-s + (−2.57 + 2.57i)23-s + (4.47 − 2.63i)27-s + 1.41·29-s − 8.99·31-s + (3.96 + 1.43i)33-s + (−7.11 + 7.11i)37-s + (−6.62 + 3.11i)39-s + ⋯
L(s)  = 1  + (−0.339 + 0.940i)3-s + (0.267 − 0.267i)7-s + (−0.769 − 0.638i)9-s − 0.733i·11-s + (0.828 + 0.828i)13-s + (−1.19 − 1.19i)17-s + 0.369i·19-s + (0.160 + 0.342i)21-s + (−0.537 + 0.537i)23-s + (0.861 − 0.507i)27-s + 0.261·29-s − 1.61·31-s + (0.690 + 0.249i)33-s + (−1.16 + 1.16i)37-s + (−1.06 + 0.498i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.546 + 0.837i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.546 + 0.837i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2100\)    =    \(2^{2} \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $-0.546 + 0.837i$
Analytic conductor: \(16.7685\)
Root analytic conductor: \(4.09494\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2100} (1793, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2100,\ (\ :1/2),\ -0.546 + 0.837i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2666506896\)
\(L(\frac12)\) \(\approx\) \(0.2666506896\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.587 - 1.62i)T \)
5 \( 1 \)
7 \( 1 + (-0.707 + 0.707i)T \)
good11 \( 1 + 2.43iT - 11T^{2} \)
13 \( 1 + (-2.98 - 2.98i)T + 13iT^{2} \)
17 \( 1 + (4.94 + 4.94i)T + 17iT^{2} \)
19 \( 1 - 1.61iT - 19T^{2} \)
23 \( 1 + (2.57 - 2.57i)T - 23iT^{2} \)
29 \( 1 - 1.41T + 29T^{2} \)
31 \( 1 + 8.99T + 31T^{2} \)
37 \( 1 + (7.11 - 7.11i)T - 37iT^{2} \)
41 \( 1 + 4.49iT - 41T^{2} \)
43 \( 1 + (4.69 + 4.69i)T + 43iT^{2} \)
47 \( 1 + (6.89 + 6.89i)T + 47iT^{2} \)
53 \( 1 + (8.78 - 8.78i)T - 53iT^{2} \)
59 \( 1 + 3.56T + 59T^{2} \)
61 \( 1 - 8.83T + 61T^{2} \)
67 \( 1 + (-6.03 + 6.03i)T - 67iT^{2} \)
71 \( 1 - 4.71iT - 71T^{2} \)
73 \( 1 + (7.52 + 7.52i)T + 73iT^{2} \)
79 \( 1 - 1.16iT - 79T^{2} \)
83 \( 1 + (2.94 - 2.94i)T - 83iT^{2} \)
89 \( 1 + 0.172T + 89T^{2} \)
97 \( 1 + (-9.38 + 9.38i)T - 97iT^{2} \)
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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

−8.874565871567798413810865419404, −8.373735741897240347885201789601, −7.13170511940497376923823079938, −6.42897895884384158316700374901, −5.53141565512251001832252107394, −4.79501344281849407578001595628, −3.90919791338004184091488038845, −3.22790337409739420328999030937, −1.76423300979280962653103499228, −0.094171033265427481633705366356, 1.52201846409040230404270020104, 2.27509333233090033615973158505, 3.53769593831704461541915564186, 4.66848936769794311189456483028, 5.53851852107222060903511939919, 6.32977574730889505031550453852, 6.91220532766069220728750423802, 7.922020399749418793893576198133, 8.387823310143382965143219528535, 9.171252688087092149708249434095

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