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Properties

Label 2-2100-105.89-c1-0-22
Degree $2$
Conductor $2100$
Sign $0.832 - 0.553i$
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  + (1.68 + 0.417i)3-s + (−2.33 + 1.25i)7-s + (2.65 + 1.40i)9-s + (4.63 − 2.67i)11-s − 4.13·13-s + (−0.134 + 0.0773i)17-s + (3.40 + 1.96i)19-s + (−4.44 + 1.13i)21-s + (2.99 − 5.19i)23-s + (3.86 + 3.46i)27-s + 10.3i·29-s + (6.70 − 3.86i)31-s + (8.91 − 2.56i)33-s + (9.07 + 5.24i)37-s + (−6.95 − 1.72i)39-s + ⋯
L(s)  = 1  + (0.970 + 0.241i)3-s + (−0.880 + 0.473i)7-s + (0.883 + 0.468i)9-s + (1.39 − 0.807i)11-s − 1.14·13-s + (−0.0325 + 0.0187i)17-s + (0.781 + 0.450i)19-s + (−0.968 + 0.247i)21-s + (0.625 − 1.08i)23-s + (0.744 + 0.667i)27-s + 1.91i·29-s + (1.20 − 0.694i)31-s + (1.55 − 0.446i)33-s + (1.49 + 0.861i)37-s + (−1.11 − 0.276i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.832 - 0.553i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.832 - 0.553i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2100\)    =    \(2^{2} \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $0.832 - 0.553i$
Analytic conductor: \(16.7685\)
Root analytic conductor: \(4.09494\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2100} (1349, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2100,\ (\ :1/2),\ 0.832 - 0.553i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.501185246\)
\(L(\frac12)\) \(\approx\) \(2.501185246\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.68 - 0.417i)T \)
5 \( 1 \)
7 \( 1 + (2.33 - 1.25i)T \)
good11 \( 1 + (-4.63 + 2.67i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + 4.13T + 13T^{2} \)
17 \( 1 + (0.134 - 0.0773i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.40 - 1.96i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2.99 + 5.19i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 10.3iT - 29T^{2} \)
31 \( 1 + (-6.70 + 3.86i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-9.07 - 5.24i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + 2.33T + 41T^{2} \)
43 \( 1 - 1.78iT - 43T^{2} \)
47 \( 1 + (3.11 + 1.80i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-2.47 - 4.29i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-5.27 - 9.12i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.25 + 1.87i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.770 + 0.444i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 11.6iT - 71T^{2} \)
73 \( 1 + (6.12 + 10.6i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (3.61 - 6.25i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 5.14iT - 83T^{2} \)
89 \( 1 + (-8.58 + 14.8i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 4.28T + 97T^{2} \)
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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

−9.034861004026344230755169724333, −8.702588987044192954876276634514, −7.68154723490766650453883297781, −6.84078381098292002370838933998, −6.20347155177923225646550205358, −5.04388547525425401865181911433, −4.14136206953628531120354938367, −3.17700274735638152039695940699, −2.64622772727715235484374402141, −1.17887391059909597655259023892, 0.944990180436216118298324989347, 2.21995145788755338963165162036, 3.14378031953714215415033722130, 4.01477161315046877593214690762, 4.73780134480395639593445890465, 6.13010909309873284227399477147, 6.97684836187280892411088615576, 7.31856327806626880514318466783, 8.202141329106332095058760744785, 9.398968197185284147216058945302

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