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Properties

Label 2-2100-15.8-c1-0-18
Degree $2$
Conductor $2100$
Sign $0.993 - 0.114i$
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  + (1.62 − 0.587i)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 + (2.63 − 4.47i)27-s − 1.41·29-s − 8.99·31-s + (1.43 + 3.96i)33-s + (−7.11 + 7.11i)37-s + (6.62 + 3.11i)39-s + ⋯
L(s)  = 1  + (0.940 − 0.339i)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 + (0.507 − 0.861i)27-s − 0.261·29-s − 1.61·31-s + (0.249 + 0.690i)33-s + (−1.16 + 1.16i)37-s + (1.06 + 0.498i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.993 - 0.114i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.993 - 0.114i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2100\)    =    \(2^{2} \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $0.993 - 0.114i$
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: \((2,\ 2100,\ (\ :1/2),\ 0.993 - 0.114i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.815068930\)
\(L(\frac12)\) \(\approx\) \(2.815068930\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.62 + 0.587i)T \)
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

−8.895053505846116219608384273241, −8.434205764110161853983920352497, −7.59122998556367220026560436981, −6.95212225468687953285291549562, −6.13971502313001080548761041376, −5.01782280691526674492079909683, −3.92172505994715596559090024899, −3.45578745431668667455564063102, −2.03143702564404906398577486613, −1.35543669091648025164705531889, 1.02941208184327659589234758074, 2.37943404376076561615964773230, 3.32894988825068345390514809512, 3.86224225258879785537843074963, 5.39085729248610036657217516325, 5.47318914740064551457716469249, 7.09713910576829156318544229555, 7.54767292344009088653650453932, 8.544796179856587160258462346975, 8.905918092811989927366318339324

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