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Properties

Label 2-2100-21.20-c1-0-23
Degree $2$
Conductor $2100$
Sign $0.924 - 0.381i$
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 − 1.41i)3-s + (2.23 + 1.41i)7-s + (−1.00 − 2.82i)9-s + 2.82i·11-s + 2.82i·13-s + 4·17-s + 6.32i·19-s + (4.23 − 1.74i)21-s + 6.32i·23-s + (−5.00 − 1.41i)27-s − 5.65i·29-s + 6.32i·31-s + (4.00 + 2.82i)33-s − 8.94·37-s + (4.00 + 2.82i)39-s + ⋯
L(s)  = 1  + (0.577 − 0.816i)3-s + (0.845 + 0.534i)7-s + (−0.333 − 0.942i)9-s + 0.852i·11-s + 0.784i·13-s + 0.970·17-s + 1.45i·19-s + (0.924 − 0.381i)21-s + 1.31i·23-s + (−0.962 − 0.272i)27-s − 1.05i·29-s + 1.13i·31-s + (0.696 + 0.492i)33-s − 1.47·37-s + (0.640 + 0.452i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.924 - 0.381i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.924 - 0.381i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2100\)    =    \(2^{2} \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $0.924 - 0.381i$
Analytic conductor: \(16.7685\)
Root analytic conductor: \(4.09494\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2100} (1301, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2100,\ (\ :1/2),\ 0.924 - 0.381i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.301704712\)
\(L(\frac12)\) \(\approx\) \(2.301704712\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1 + 1.41i)T \)
5 \( 1 \)
7 \( 1 + (-2.23 - 1.41i)T \)
good11 \( 1 - 2.82iT - 11T^{2} \)
13 \( 1 - 2.82iT - 13T^{2} \)
17 \( 1 - 4T + 17T^{2} \)
19 \( 1 - 6.32iT - 19T^{2} \)
23 \( 1 - 6.32iT - 23T^{2} \)
29 \( 1 + 5.65iT - 29T^{2} \)
31 \( 1 - 6.32iT - 31T^{2} \)
37 \( 1 + 8.94T + 37T^{2} \)
41 \( 1 - 4.47T + 41T^{2} \)
43 \( 1 + 4.47T + 43T^{2} \)
47 \( 1 - 6T + 47T^{2} \)
53 \( 1 - 6.32iT - 53T^{2} \)
59 \( 1 - 8.94T + 59T^{2} \)
61 \( 1 - 61T^{2} \)
67 \( 1 + 4.47T + 67T^{2} \)
71 \( 1 + 14.1iT - 71T^{2} \)
73 \( 1 + 8.48iT - 73T^{2} \)
79 \( 1 - 12T + 79T^{2} \)
83 \( 1 + 10T + 83T^{2} \)
89 \( 1 - 4.47T + 89T^{2} \)
97 \( 1 - 8.48iT - 97T^{2} \)
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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

−9.079217782041162630606107393556, −8.227026800344871139047783934524, −7.67138475525927827978425108689, −7.03309527065081919644737195769, −5.99988915427465880613300077785, −5.29876926691340120909129357250, −4.15344920555672755489392341984, −3.23489457525188049906618901211, −1.97223543487132668589267210748, −1.48081326345212060102062957224, 0.790199277416321031072224109691, 2.36812941506272883274328947823, 3.26764716892190751106070420275, 4.10647763663342595868524236750, 5.03699694359572126519009859882, 5.55932277456644856315615244037, 6.86939179838807120562632158849, 7.69779302753121983625907682364, 8.452234045463182103658425189431, 8.842943755439413920096953322772

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