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Properties

Label 2-2100-15.8-c1-0-29
Degree $2$
Conductor $2100$
Sign $-0.484 + 0.874i$
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  + (1.28 − 1.15i)3-s + (0.707 − 0.707i)7-s + (0.317 − 2.98i)9-s − 1.81i·11-s + (−2.28 − 2.28i)13-s + (0.614 + 0.614i)17-s + 0.915i·19-s + (0.0918 − 1.72i)21-s + (3.11 − 3.11i)23-s + (−3.04 − 4.21i)27-s − 6.83·29-s + 8.15·31-s + (−2.10 − 2.33i)33-s + (−1.57 + 1.57i)37-s + (−5.59 − 0.297i)39-s + ⋯
L(s)  = 1  + (0.743 − 0.668i)3-s + (0.267 − 0.267i)7-s + (0.105 − 0.994i)9-s − 0.547i·11-s + (−0.634 − 0.634i)13-s + (0.148 + 0.148i)17-s + 0.210i·19-s + (0.0200 − 0.377i)21-s + (0.649 − 0.649i)23-s + (−0.586 − 0.810i)27-s − 1.26·29-s + 1.46·31-s + (−0.366 − 0.407i)33-s + (−0.259 + 0.259i)37-s + (−0.895 − 0.0475i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.484 + 0.874i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.484 + 0.874i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2100\)    =    \(2^{2} \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $-0.484 + 0.874i$
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: \((2,\ 2100,\ (\ :1/2),\ -0.484 + 0.874i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.999917882\)
\(L(\frac12)\) \(\approx\) \(1.999917882\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.28 + 1.15i)T \)
5 \( 1 \)
7 \( 1 + (-0.707 + 0.707i)T \)
good11 \( 1 + 1.81iT - 11T^{2} \)
13 \( 1 + (2.28 + 2.28i)T + 13iT^{2} \)
17 \( 1 + (-0.614 - 0.614i)T + 17iT^{2} \)
19 \( 1 - 0.915iT - 19T^{2} \)
23 \( 1 + (-3.11 + 3.11i)T - 23iT^{2} \)
29 \( 1 + 6.83T + 29T^{2} \)
31 \( 1 - 8.15T + 31T^{2} \)
37 \( 1 + (1.57 - 1.57i)T - 37iT^{2} \)
41 \( 1 + 0.926iT - 41T^{2} \)
43 \( 1 + (3.34 + 3.34i)T + 43iT^{2} \)
47 \( 1 + (7.74 + 7.74i)T + 47iT^{2} \)
53 \( 1 + (-3.13 + 3.13i)T - 53iT^{2} \)
59 \( 1 + 9.50T + 59T^{2} \)
61 \( 1 - 0.778T + 61T^{2} \)
67 \( 1 + (-3.95 + 3.95i)T - 67iT^{2} \)
71 \( 1 - 10.0iT - 71T^{2} \)
73 \( 1 + (-3.51 - 3.51i)T + 73iT^{2} \)
79 \( 1 - 8.66iT - 79T^{2} \)
83 \( 1 + (-9.57 + 9.57i)T - 83iT^{2} \)
89 \( 1 + 8.12T + 89T^{2} \)
97 \( 1 + (-3.33 + 3.33i)T - 97iT^{2} \)
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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

−8.549883770548831900072861611421, −8.204807748210630573609151188275, −7.33753816290908166565358136411, −6.69570978009201415780527269197, −5.75729412737619717702992545622, −4.79986205253819834935875530708, −3.67094569126364858211956633291, −2.91002913773791533641706303610, −1.86624268751915826223150833977, −0.62242862355659411551258900853, 1.67896959648875817097991421778, 2.62754469400322406332120628855, 3.55050967551308142822519544898, 4.63000133595217569491099606715, 5.01799521553554180787520302479, 6.21085477591306008960501422914, 7.28248426354989730827089567727, 7.81274886831329010499749110512, 8.714431391547736793411665041036, 9.463282996964653198640303472141

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