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Properties

Label 2-2100-7.2-c1-0-11
Degree $2$
Conductor $2100$
Sign $-0.0633 - 0.997i$
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 + (1.32 + 2.29i)7-s + (−0.499 + 0.866i)9-s + (2.82 + 4.88i)11-s + 13-s + (0.822 + 1.42i)17-s + (2.32 − 4.02i)19-s + (−1.32 + 2.29i)21-s + (1 − 1.73i)23-s − 0.999·27-s + 7.29·29-s + (−2 − 3.46i)31-s + (−2.82 + 4.88i)33-s + (−4.14 + 7.18i)37-s + (0.5 + 0.866i)39-s + ⋯
L(s)  = 1  + (0.288 + 0.499i)3-s + (0.499 + 0.866i)7-s + (−0.166 + 0.288i)9-s + (0.851 + 1.47i)11-s + 0.277·13-s + (0.199 + 0.345i)17-s + (0.532 − 0.923i)19-s + (−0.288 + 0.499i)21-s + (0.208 − 0.361i)23-s − 0.192·27-s + 1.35·29-s + (−0.359 − 0.622i)31-s + (−0.491 + 0.851i)33-s + (−0.681 + 1.18i)37-s + (0.0800 + 0.138i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0633 - 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0633 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2100\)    =    \(2^{2} \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $-0.0633 - 0.997i$
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: \((2,\ 2100,\ (\ :1/2),\ -0.0633 - 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.175955576\)
\(L(\frac12)\) \(\approx\) \(2.175955576\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.32 - 2.29i)T \)
good11 \( 1 + (-2.82 - 4.88i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - T + 13T^{2} \)
17 \( 1 + (-0.822 - 1.42i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.32 + 4.02i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1 + 1.73i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 7.29T + 29T^{2} \)
31 \( 1 + (2 + 3.46i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (4.14 - 7.18i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 0.354T + 41T^{2} \)
43 \( 1 + 7.29T + 43T^{2} \)
47 \( 1 + (1.46 - 2.54i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (1.17 + 2.03i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (7.46 + 12.9i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (4.79 - 8.29i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.96 - 6.87i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 3.29T + 71T^{2} \)
73 \( 1 + (-3.14 - 5.44i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-3.32 + 5.75i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 3.64T + 83T^{2} \)
89 \( 1 + (-4.46 + 7.73i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 14.2T + 97T^{2} \)
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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

−9.317143771520546109493476942849, −8.624972289570730510693052869746, −7.920092195088720830731235046959, −6.92294295077234287207282401486, −6.22335607433833516580176663366, −4.97265109191210842081068454561, −4.67836377980628206571470607949, −3.51880662395650570328080120776, −2.48772151417875532885191817333, −1.50001483961399123400964042113, 0.810022429285912094020162092029, 1.65339262427500792761075656774, 3.23289565860442258193798858965, 3.69478294487846053654327893406, 4.88993037165904390080933231280, 5.86124544058263150574532025420, 6.61409711183414296675320110360, 7.39751658546863938511355089709, 8.161720533981055117264786430490, 8.725518872847310518195981526048

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