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Properties

Label 2-2100-105.104-c1-0-11
Degree $2$
Conductor $2100$
Sign $0.716 - 0.697i$
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  + (0.437 − 1.67i)3-s + (1.41 + 2.23i)7-s + (−2.61 − 1.46i)9-s + 3.83i·11-s − 5.99·13-s + 2.07i·17-s − 5.11i·19-s + (4.36 − 1.39i)21-s + 8.57·23-s + (−3.59 + 3.74i)27-s + 5.64i·29-s − 1.95i·31-s + (6.42 + 1.67i)33-s + 2.23i·37-s + (−2.61 + 10.0i)39-s + ⋯
L(s)  = 1  + (0.252 − 0.967i)3-s + (0.534 + 0.845i)7-s + (−0.872 − 0.488i)9-s + 1.15i·11-s − 1.66·13-s + 0.502i·17-s − 1.17i·19-s + (0.952 − 0.303i)21-s + 1.78·23-s + (−0.692 + 0.721i)27-s + 1.04i·29-s − 0.351i·31-s + (1.11 + 0.291i)33-s + 0.367i·37-s + (−0.419 + 1.60i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.716 - 0.697i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.716 - 0.697i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2100\)    =    \(2^{2} \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $0.716 - 0.697i$
Analytic conductor: \(16.7685\)
Root analytic conductor: \(4.09494\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2100} (1049, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2100,\ (\ :1/2),\ 0.716 - 0.697i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.520209268\)
\(L(\frac12)\) \(\approx\) \(1.520209268\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.437 + 1.67i)T \)
5 \( 1 \)
7 \( 1 + (-1.41 - 2.23i)T \)
good11 \( 1 - 3.83iT - 11T^{2} \)
13 \( 1 + 5.99T + 13T^{2} \)
17 \( 1 - 2.07iT - 17T^{2} \)
19 \( 1 + 5.11iT - 19T^{2} \)
23 \( 1 - 8.57T + 23T^{2} \)
29 \( 1 - 5.64iT - 29T^{2} \)
31 \( 1 + 1.95iT - 31T^{2} \)
37 \( 1 - 2.23iT - 37T^{2} \)
41 \( 1 - 7.49T + 41T^{2} \)
43 \( 1 - 9.47iT - 43T^{2} \)
47 \( 1 - 12.9iT - 47T^{2} \)
53 \( 1 + 10.6T + 53T^{2} \)
59 \( 1 - 12.1T + 59T^{2} \)
61 \( 1 - 4.37iT - 61T^{2} \)
67 \( 1 - 6.70iT - 67T^{2} \)
71 \( 1 + 2.02iT - 71T^{2} \)
73 \( 1 - 1.41T + 73T^{2} \)
79 \( 1 - 7.47T + 79T^{2} \)
83 \( 1 + 7.49iT - 83T^{2} \)
89 \( 1 - 2.86T + 89T^{2} \)
97 \( 1 - 0.206T + 97T^{2} \)
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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

−9.257240628386701760741072100579, −8.351553106858117930173127916290, −7.47787897752458001631834260479, −7.10119308570852589070536799945, −6.19033504069320990103599258816, −5.07146611298732637578379667870, −4.65170152747789446647755824166, −2.88599620173294739358843606842, −2.41385979423064451870437758610, −1.29607572722679199767805864398, 0.53502261239925637779723489293, 2.27396748441911710114949009966, 3.29706111130732549728709534015, 4.06802094474283930295627619900, 5.02556126902243307900381159344, 5.47606905384431784994136649240, 6.76422884509831558385970547602, 7.61637005070625175912181598854, 8.248303477409519697693128486729, 9.098328213645495316800627696801

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