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Properties

Label 2-2100-15.2-c1-0-3
Degree $2$
Conductor $2100$
Sign $-0.993 + 0.109i$
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  + (−1.71 − 0.210i)3-s + (0.707 + 0.707i)7-s + (2.91 + 0.725i)9-s + 6.43i·11-s + (−4.23 + 4.23i)13-s + (−1.00 + 1.00i)17-s − 6.08i·19-s + (−1.06 − 1.36i)21-s + (0.649 + 0.649i)23-s + (−4.85 − 1.86i)27-s − 0.486·29-s − 2.38·31-s + (1.35 − 11.0i)33-s + (−4.10 − 4.10i)37-s + (8.17 − 6.39i)39-s + ⋯
L(s)  = 1  + (−0.992 − 0.121i)3-s + (0.267 + 0.267i)7-s + (0.970 + 0.241i)9-s + 1.93i·11-s + (−1.17 + 1.17i)13-s + (−0.243 + 0.243i)17-s − 1.39i·19-s + (−0.232 − 0.297i)21-s + (0.135 + 0.135i)23-s + (−0.933 − 0.358i)27-s − 0.0902·29-s − 0.427·31-s + (0.236 − 1.92i)33-s + (−0.674 − 0.674i)37-s + (1.30 − 1.02i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.993 + 0.109i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.993 + 0.109i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2100\)    =    \(2^{2} \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $-0.993 + 0.109i$
Analytic conductor: \(16.7685\)
Root analytic conductor: \(4.09494\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2100} (1457, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2100,\ (\ :1/2),\ -0.993 + 0.109i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2836924551\)
\(L(\frac12)\) \(\approx\) \(0.2836924551\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.71 + 0.210i)T \)
5 \( 1 \)
7 \( 1 + (-0.707 - 0.707i)T \)
good11 \( 1 - 6.43iT - 11T^{2} \)
13 \( 1 + (4.23 - 4.23i)T - 13iT^{2} \)
17 \( 1 + (1.00 - 1.00i)T - 17iT^{2} \)
19 \( 1 + 6.08iT - 19T^{2} \)
23 \( 1 + (-0.649 - 0.649i)T + 23iT^{2} \)
29 \( 1 + 0.486T + 29T^{2} \)
31 \( 1 + 2.38T + 31T^{2} \)
37 \( 1 + (4.10 + 4.10i)T + 37iT^{2} \)
41 \( 1 - 5.91iT - 41T^{2} \)
43 \( 1 + (-6.74 + 6.74i)T - 43iT^{2} \)
47 \( 1 + (-4.70 + 4.70i)T - 47iT^{2} \)
53 \( 1 + (-4.00 - 4.00i)T + 53iT^{2} \)
59 \( 1 + 8.86T + 59T^{2} \)
61 \( 1 + 7.72T + 61T^{2} \)
67 \( 1 + (8.66 + 8.66i)T + 67iT^{2} \)
71 \( 1 + 0.938iT - 71T^{2} \)
73 \( 1 + (-0.399 + 0.399i)T - 73iT^{2} \)
79 \( 1 - 6.65iT - 79T^{2} \)
83 \( 1 + (8.35 + 8.35i)T + 83iT^{2} \)
89 \( 1 + 8.32T + 89T^{2} \)
97 \( 1 + (-1.26 - 1.26i)T + 97iT^{2} \)
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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

−9.437716093928802785670846762741, −9.054886292983944131956276744956, −7.47312969130266761742093742591, −7.22884097584289658734692794827, −6.51895343831101365101570311923, −5.39170405582684751432438657367, −4.66259058518019643806314322510, −4.26243211727619862538704987734, −2.40932275464449748563059614383, −1.70107429925208722190475395708, 0.12092188620489955268635796539, 1.23318299247757131282524329751, 2.85807057280734945808804744948, 3.79500188191957289650036014577, 4.81259401771094950813400400180, 5.69846078150769034226119267740, 5.99523543024639966328415760089, 7.18631632869923385326481271381, 7.83185229566742833179940737333, 8.645787129956843625660628609263

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