↑

Properties

Label 2-2100-35.17-c1-0-23
Degree $2$
Conductor $2100$
Sign $-0.683 + 0.730i$
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 + (0.588 − 2.57i)7-s + (0.866 − 0.499i)9-s + (−0.401 + 0.694i)11-s + (−2.60 − 2.60i)13-s + (−0.207 − 0.774i)17-s + (−1.40 − 2.43i)19-s + (−0.0989 − 2.64i)21-s + (−8.12 − 2.17i)23-s + (0.707 − 0.707i)27-s + (−5.14 − 2.96i)31-s + (−0.207 + 0.774i)33-s + (−0.448 + 1.67i)37-s + (−3.18 − 1.83i)39-s + 7.01i·41-s + ⋯
L(s)  = 1  + (0.557 − 0.149i)3-s + (0.222 − 0.974i)7-s + (0.288 − 0.166i)9-s + (−0.120 + 0.209i)11-s + (−0.721 − 0.721i)13-s + (−0.0503 − 0.187i)17-s + (−0.322 − 0.559i)19-s + (−0.0215 − 0.576i)21-s + (−1.69 − 0.453i)23-s + (0.136 − 0.136i)27-s + (−0.923 − 0.533i)31-s + (−0.0361 + 0.134i)33-s + (−0.0736 + 0.275i)37-s + (−0.510 − 0.294i)39-s + 1.09i·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.683 + 0.730i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.683 + 0.730i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2100\)    =    \(2^{2} \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $-0.683 + 0.730i$
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: \((2,\ 2100,\ (\ :1/2),\ -0.683 + 0.730i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.337356632\)
\(L(\frac12)\) \(\approx\) \(1.337356632\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.588 + 2.57i)T \)
good11 \( 1 + (0.401 - 0.694i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (2.60 + 2.60i)T + 13iT^{2} \)
17 \( 1 + (0.207 + 0.774i)T + (-14.7 + 8.5i)T^{2} \)
19 \( 1 + (1.40 + 2.43i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (8.12 + 2.17i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 - 29T^{2} \)
31 \( 1 + (5.14 + 2.96i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.448 - 1.67i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 - 7.01iT - 41T^{2} \)
43 \( 1 + (-2.23 + 2.23i)T - 43iT^{2} \)
47 \( 1 + (-10.1 - 2.73i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (3.27 + 12.2i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (3.50 - 6.07i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-8.78 + 5.07i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.71 + 0.728i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + 5.35T + 71T^{2} \)
73 \( 1 + (7.68 - 2.06i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (5.85 - 3.38i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (0.112 + 0.112i)T + 83iT^{2} \)
89 \( 1 + (-0.694 - 1.20i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-3.53 + 3.53i)T - 97iT^{2} \)
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

−8.703464026112904371435172994298, −7.929272571207594287915951597987, −7.42736317335612503315393954892, −6.65679596621660657854472299547, −5.62281701613198146268534044920, −4.58674426249663383724816017139, −3.92753928274249166470407124265, −2.83358470515269862867753705242, −1.87061172354056922795814064233, −0.40213866519872833956183391694, 1.80188512032368419731521460948, 2.45961966607214613295059111544, 3.64478802475347469977424500317, 4.43998081379610203110582406481, 5.49694125075757192084587137790, 6.10459946160280169743629546894, 7.25165949299314142244304964933, 7.88614018586187056779821591216, 8.769457789812500436336175795876, 9.188108844922635336198785720885

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