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Properties

Label 2-2100-7.2-c1-0-13
Degree $2$
Conductor $2100$
Sign $0.954 + 0.296i$
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  + (−0.5 − 0.866i)3-s + (2.61 − 0.418i)7-s + (−0.499 + 0.866i)9-s + (0.292 + 0.506i)11-s − 1.75·13-s + (3.11 + 5.39i)17-s + (3.48 − 6.02i)19-s + (−1.66 − 2.05i)21-s + (−1.37 + 2.38i)23-s + 0.999·27-s + 3.24·29-s + (1.25 + 2.16i)31-s + (0.292 − 0.506i)33-s + (−1.94 + 3.36i)37-s + (0.875 + 1.51i)39-s + ⋯
L(s)  = 1  + (−0.288 − 0.499i)3-s + (0.987 − 0.158i)7-s + (−0.166 + 0.288i)9-s + (0.0881 + 0.152i)11-s − 0.485·13-s + (0.754 + 1.30i)17-s + (0.798 − 1.38i)19-s + (−0.364 − 0.448i)21-s + (−0.286 + 0.496i)23-s + 0.192·27-s + 0.601·29-s + (0.224 + 0.389i)31-s + (0.0508 − 0.0881i)33-s + (−0.319 + 0.553i)37-s + (0.140 + 0.242i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.954 + 0.296i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.954 + 0.296i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2100\)    =    \(2^{2} \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $0.954 + 0.296i$
Analytic conductor: \(16.7685\)
Root analytic conductor: \(4.09494\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2100} (1801, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2100,\ (\ :1/2),\ 0.954 + 0.296i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.829574089\)
\(L(\frac12)\) \(\approx\) \(1.829574089\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.5 + 0.866i)T \)
5 \( 1 \)
7 \( 1 + (-2.61 + 0.418i)T \)
good11 \( 1 + (-0.292 - 0.506i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 1.75T + 13T^{2} \)
17 \( 1 + (-3.11 - 5.39i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3.48 + 6.02i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.37 - 2.38i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 3.24T + 29T^{2} \)
31 \( 1 + (-1.25 - 2.16i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (1.94 - 3.36i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 4.58T + 41T^{2} \)
43 \( 1 + 0.754T + 43T^{2} \)
47 \( 1 + (-0.727 + 1.26i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (0.263 + 0.456i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-5.05 - 8.75i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (4.07 - 7.06i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (7.34 + 12.7i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 13.3T + 71T^{2} \)
73 \( 1 + (-3.37 - 5.84i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-4.04 + 7.01i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 5.72T + 83T^{2} \)
89 \( 1 + (-8.50 + 14.7i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 11.1T + 97T^{2} \)
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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

−8.942374097094890145038673298368, −8.155114731007193452606529604219, −7.53200237508001295013081387020, −6.84310589569918367223323308218, −5.85646878262939085668989489786, −5.09682767153326858441274622800, −4.34386006421650134708215764765, −3.12692008926710381486927964111, −1.96207916337746015270048296656, −0.981808182559456533830578153990, 0.918020269381058049675684272833, 2.28028614132310426734802559159, 3.39189986697798024997674546928, 4.37910134096577901592656754666, 5.19236144171898227066141510309, 5.71056937577261548800192650415, 6.81739100459791572632236620978, 7.77429831625412420826165868037, 8.216917494016191533194964404444, 9.347874102485816066219410874004

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