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Properties

Label 2-2100-15.8-c1-0-10
Degree $2$
Conductor $2100$
Sign $0.993 - 0.117i$
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  + (−1.69 − 0.369i)3-s + (0.707 − 0.707i)7-s + (2.72 + 1.25i)9-s − 4.03i·11-s + (3.70 + 3.70i)13-s + (5.26 + 5.26i)17-s − 2.18i·19-s + (−1.45 + 0.935i)21-s + (−3.73 + 3.73i)23-s + (−4.15 − 3.12i)27-s − 7.12·29-s − 3.41·31-s + (−1.49 + 6.83i)33-s + (4.40 − 4.40i)37-s + (−4.89 − 7.63i)39-s + ⋯
L(s)  = 1  + (−0.976 − 0.213i)3-s + (0.267 − 0.267i)7-s + (0.908 + 0.417i)9-s − 1.21i·11-s + (1.02 + 1.02i)13-s + (1.27 + 1.27i)17-s − 0.500i·19-s + (−0.318 + 0.204i)21-s + (−0.778 + 0.778i)23-s + (−0.798 − 0.601i)27-s − 1.32·29-s − 0.613·31-s + (−0.259 + 1.18i)33-s + (0.724 − 0.724i)37-s + (−0.783 − 1.22i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.993 - 0.117i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.993 - 0.117i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2100\)    =    \(2^{2} \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $0.993 - 0.117i$
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: \((2,\ 2100,\ (\ :1/2),\ 0.993 - 0.117i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.351972661\)
\(L(\frac12)\) \(\approx\) \(1.351972661\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.69 + 0.369i)T \)
5 \( 1 \)
7 \( 1 + (-0.707 + 0.707i)T \)
good11 \( 1 + 4.03iT - 11T^{2} \)
13 \( 1 + (-3.70 - 3.70i)T + 13iT^{2} \)
17 \( 1 + (-5.26 - 5.26i)T + 17iT^{2} \)
19 \( 1 + 2.18iT - 19T^{2} \)
23 \( 1 + (3.73 - 3.73i)T - 23iT^{2} \)
29 \( 1 + 7.12T + 29T^{2} \)
31 \( 1 + 3.41T + 31T^{2} \)
37 \( 1 + (-4.40 + 4.40i)T - 37iT^{2} \)
41 \( 1 - 0.501iT - 41T^{2} \)
43 \( 1 + (-8.62 - 8.62i)T + 43iT^{2} \)
47 \( 1 + (5.51 + 5.51i)T + 47iT^{2} \)
53 \( 1 + (5.79 - 5.79i)T - 53iT^{2} \)
59 \( 1 - 13.0T + 59T^{2} \)
61 \( 1 - 12.3T + 61T^{2} \)
67 \( 1 + (-5.56 + 5.56i)T - 67iT^{2} \)
71 \( 1 - 11.2iT - 71T^{2} \)
73 \( 1 + (-0.731 - 0.731i)T + 73iT^{2} \)
79 \( 1 + 3.73iT - 79T^{2} \)
83 \( 1 + (1.07 - 1.07i)T - 83iT^{2} \)
89 \( 1 - 8.06T + 89T^{2} \)
97 \( 1 + (-1.14 + 1.14i)T - 97iT^{2} \)
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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

−9.155682602260130859892689969234, −8.189114774184539761005237025613, −7.59560184671979369114827124162, −6.59104952574121339100145162015, −5.87634913697225591102582228195, −5.44882959866584619634077949500, −4.09369834308220003693383871756, −3.62421967515565540219375627825, −1.86846756340984018774974144900, −0.941251202760164021629533929666, 0.75103139956923832258093230011, 2.00906034078477541521401558921, 3.42485265700976787217336484468, 4.30115612028123889139159597857, 5.32820313990197224358882483718, 5.66826048562301477510684286603, 6.69417871426446716692028438359, 7.51861338141877729838070573970, 8.147828650206111181311673354638, 9.352098889473944008295644235294

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