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Properties

Label 2-2100-21.17-c1-0-39
Degree $2$
Conductor $2100$
Sign $0.389 + 0.921i$
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  + (1.52 − 0.813i)3-s + (0.567 − 2.58i)7-s + (1.67 − 2.48i)9-s + (0.793 + 0.457i)11-s + 4.31i·13-s + (−0.268 + 0.465i)17-s + (1.12 − 0.651i)19-s + (−1.23 − 4.41i)21-s + (6.91 − 3.99i)23-s + (0.537 − 5.16i)27-s − 3.46i·29-s + (−5.56 − 3.21i)31-s + (1.58 + 0.0547i)33-s + (2.52 + 4.36i)37-s + (3.50 + 6.59i)39-s + ⋯
L(s)  = 1  + (0.882 − 0.469i)3-s + (0.214 − 0.976i)7-s + (0.558 − 0.829i)9-s + (0.239 + 0.138i)11-s + 1.19i·13-s + (−0.0651 + 0.112i)17-s + (0.258 − 0.149i)19-s + (−0.269 − 0.962i)21-s + (1.44 − 0.832i)23-s + (0.103 − 0.994i)27-s − 0.642i·29-s + (−0.998 − 0.576i)31-s + (0.275 + 0.00952i)33-s + (0.414 + 0.718i)37-s + (0.561 + 1.05i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.389 + 0.921i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.389 + 0.921i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2100\)    =    \(2^{2} \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $0.389 + 0.921i$
Analytic conductor: \(16.7685\)
Root analytic conductor: \(4.09494\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2100} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2100,\ (\ :1/2),\ 0.389 + 0.921i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.607480845\)
\(L(\frac12)\) \(\approx\) \(2.607480845\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.52 + 0.813i)T \)
5 \( 1 \)
7 \( 1 + (-0.567 + 2.58i)T \)
good11 \( 1 + (-0.793 - 0.457i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 - 4.31iT - 13T^{2} \)
17 \( 1 + (0.268 - 0.465i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.12 + 0.651i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-6.91 + 3.99i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 3.46iT - 29T^{2} \)
31 \( 1 + (5.56 + 3.21i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-2.52 - 4.36i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 9.89T + 41T^{2} \)
43 \( 1 - 8.22T + 43T^{2} \)
47 \( 1 + (2.17 + 3.75i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (1.70 + 0.984i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-7.15 + 12.3i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (8.38 - 4.84i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (6.05 - 10.4i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 0.943iT - 71T^{2} \)
73 \( 1 + (8.85 + 5.11i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (5.71 + 9.90i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 9.35T + 83T^{2} \)
89 \( 1 + (0.874 + 1.51i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 10.7iT - 97T^{2} \)
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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

−9.039063086321316738880726217990, −8.139140426831615438589997288715, −7.31050223732953755530733886368, −6.92637622962838113144106816501, −6.02299284925090959413826417814, −4.55821231599612821732708804812, −4.10395923915255369154892953891, −3.02357429980504746708255238273, −1.97146723813279180942128462721, −0.911484123271641262421361131004, 1.39878915830785062104838074995, 2.70044642942816070062008935968, 3.20704119513151932826341625560, 4.32744321994424721955934325643, 5.31485604219546185772161211549, 5.81669250675959679084933126510, 7.23552914538521446323961221704, 7.73003428288097538365120997974, 8.692929698478133719866522840935, 9.122888547810050539980135538796

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