Properties

Degree $2$
Conductor $2100$
Sign $0.947 - 0.318i$
Motivic weight $1$
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 + (2.62 + 0.358i)7-s + (−0.499 − 0.866i)9-s + (−2.12 + 3.67i)11-s − 5·13-s + (2.12 − 3.67i)17-s + (2.62 + 4.54i)19-s + (1.62 − 2.09i)21-s + (3 + 5.19i)23-s − 0.999·27-s + 8.48·29-s + (2 − 3.46i)31-s + (2.12 + 3.67i)33-s + (2.5 + 4.33i)37-s + (−2.5 + 4.33i)39-s + ⋯
L(s)  = 1  + (0.288 − 0.499i)3-s + (0.990 + 0.135i)7-s + (−0.166 − 0.288i)9-s + (−0.639 + 1.10i)11-s − 1.38·13-s + (0.514 − 0.891i)17-s + (0.601 + 1.04i)19-s + (0.353 − 0.456i)21-s + (0.625 + 1.08i)23-s − 0.192·27-s + 1.57·29-s + (0.359 − 0.622i)31-s + (0.369 + 0.639i)33-s + (0.410 + 0.711i)37-s + (−0.400 + 0.693i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.947 - 0.318i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.947 - 0.318i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2100\)    =    \(2^{2} \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $0.947 - 0.318i$
Motivic weight: \(1\)
Character: $\chi_{2100} (1201, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2100,\ (\ :1/2),\ 0.947 - 0.318i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.014639388\)
\(L(\frac12)\) \(\approx\) \(2.014639388\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.62 - 0.358i)T \)
good11 \( 1 + (2.12 - 3.67i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 5T + 13T^{2} \)
17 \( 1 + (-2.12 + 3.67i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.62 - 4.54i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-3 - 5.19i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 8.48T + 29T^{2} \)
31 \( 1 + (-2 + 3.46i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2.5 - 4.33i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 10.2T + 41T^{2} \)
43 \( 1 - 12.4T + 43T^{2} \)
47 \( 1 + (-5.12 - 8.87i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-2.12 + 3.67i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (5.12 - 8.87i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.5 - 0.866i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-5.86 + 10.1i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 8.48T + 71T^{2} \)
73 \( 1 + (1.74 - 3.01i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (4.62 + 8.00i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 1.75T + 83T^{2} \)
89 \( 1 + (3.87 + 6.71i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 17.9T + 97T^{2} \)
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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

−9.187368700411127643826981046351, −8.122315117619759550254675461790, −7.48748814382932298993075374696, −7.25683774994983399652482079424, −5.90786669397225819413715186453, −5.02425844198564675425156190050, −4.52246026204136968592771709365, −3.02716093617368448069537749681, −2.27130608841720547962431179718, −1.18145058467931711688670654678, 0.78422482273165151599192339609, 2.38599113892756873831395510877, 3.08068306119921051674535457002, 4.32749048261804093995022526424, 4.99008084440527126610594509979, 5.64082083506837630840228247212, 6.86070152816569577953877464233, 7.63730391764069203448661659607, 8.446424111374544381050895478924, 8.824334937984847925660164834529

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