↑

Properties

Label 2-2100-35.17-c1-0-17
Degree $2$
Conductor $2100$
Sign $0.247 + 0.968i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.965 − 0.258i)3-s + (2.42 − 1.06i)7-s + (0.866 − 0.499i)9-s + (−1.81 + 3.13i)11-s + (−4.28 − 4.28i)13-s + (−1.60 − 5.99i)17-s + (0.974 + 1.68i)19-s + (2.06 − 1.65i)21-s + (5.55 + 1.48i)23-s + (0.707 − 0.707i)27-s − 6.10i·29-s + (1.14 + 0.659i)31-s + (−0.937 + 3.49i)33-s + (3.08 − 11.5i)37-s + (−5.24 − 3.02i)39-s + ⋯
L(s)  = 1  + (0.557 − 0.149i)3-s + (0.915 − 0.403i)7-s + (0.288 − 0.166i)9-s + (−0.545 + 0.945i)11-s + (−1.18 − 1.18i)13-s + (−0.389 − 1.45i)17-s + (0.223 + 0.387i)19-s + (0.450 − 0.361i)21-s + (1.15 + 0.310i)23-s + (0.136 − 0.136i)27-s − 1.13i·29-s + (0.205 + 0.118i)31-s + (−0.163 + 0.608i)33-s + (0.507 − 1.89i)37-s + (−0.840 − 0.485i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.247 + 0.968i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.247 + 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2100\)    =    \(2^{2} \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $0.247 + 0.968i$
Analytic conductor: \(16.7685\)
Root analytic conductor: \(4.09494\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2100} (157, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2100,\ (\ :1/2),\ 0.247 + 0.968i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.036598426\)
\(L(\frac12)\) \(\approx\) \(2.036598426\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.965 + 0.258i)T \)
5 \( 1 \)
7 \( 1 + (-2.42 + 1.06i)T \)
good11 \( 1 + (1.81 - 3.13i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (4.28 + 4.28i)T + 13iT^{2} \)
17 \( 1 + (1.60 + 5.99i)T + (-14.7 + 8.5i)T^{2} \)
19 \( 1 + (-0.974 - 1.68i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-5.55 - 1.48i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 + 6.10iT - 29T^{2} \)
31 \( 1 + (-1.14 - 0.659i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-3.08 + 11.5i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + 4.50iT - 41T^{2} \)
43 \( 1 + (-6.04 + 6.04i)T - 43iT^{2} \)
47 \( 1 + (7.81 + 2.09i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (-2.59 - 9.67i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (3.43 - 5.94i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (5.74 - 3.31i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.08 + 0.825i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 - 1.00T + 71T^{2} \)
73 \( 1 + (-2.86 + 0.767i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (-10.9 + 6.31i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-4.26 - 4.26i)T + 83iT^{2} \)
89 \( 1 + (-5.82 - 10.0i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (8.89 - 8.89i)T - 97iT^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.043012582800689206796021726511, −7.79802217660385671252439130803, −7.60942743527982131705455169489, −6.98726763523607054038196271465, −5.48699641308503670743468976157, −4.95022608534513063136688483461, −4.10486343239883679056145837355, −2.79554458865754539177163714392, −2.18293277765005058823806646466, −0.67399568795866130241444245481, 1.44964987964992433028852329733, 2.46219524658058056136640521742, 3.33726087854495210665488786083, 4.64904628129358157663092811741, 4.94248996551647757956265858251, 6.19637624342010695330361901735, 6.96202328555570165323996179842, 8.007206916026491834412259718221, 8.398369714511625182382833200670, 9.173472135758534032862754056231

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