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Properties

Label 2-2100-5.4-c3-0-47
Degree $2$
Conductor $2100$
Sign $-0.894 + 0.447i$
Analytic cond. $123.904$
Root an. cond. $11.1312$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3i·3-s − 7i·7-s − 9·9-s + 6.11·11-s − 15.2i·13-s − 101. i·17-s + 72.9·19-s − 21·21-s − 138. i·23-s + 27i·27-s − 71.7·29-s + 212.·31-s − 18.3i·33-s + 66.9i·37-s − 45.6·39-s + ⋯
L(s)  = 1  − 0.577i·3-s − 0.377i·7-s − 0.333·9-s + 0.167·11-s − 0.324i·13-s − 1.44i·17-s + 0.880·19-s − 0.218·21-s − 1.25i·23-s + 0.192i·27-s − 0.459·29-s + 1.23·31-s − 0.0968i·33-s + 0.297i·37-s − 0.187·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2100\)    =    \(2^{2} \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $-0.894 + 0.447i$
Analytic conductor: \(123.904\)
Root analytic conductor: \(11.1312\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{2100} (1849, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2100,\ (\ :3/2),\ -0.894 + 0.447i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.650695555\)
\(L(\frac12)\) \(\approx\) \(1.650695555\)
\(L(\frac{5}{2})\) 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 \( 1 + 3iT \)
5 \( 1 \)
7 \( 1 + 7iT \)
good11 \( 1 - 6.11T + 1.33e3T^{2} \)
13 \( 1 + 15.2iT - 2.19e3T^{2} \)
17 \( 1 + 101. iT - 4.91e3T^{2} \)
19 \( 1 - 72.9T + 6.85e3T^{2} \)
23 \( 1 + 138. iT - 1.21e4T^{2} \)
29 \( 1 + 71.7T + 2.43e4T^{2} \)
31 \( 1 - 212.T + 2.97e4T^{2} \)
37 \( 1 - 66.9iT - 5.06e4T^{2} \)
41 \( 1 + 12.4T + 6.89e4T^{2} \)
43 \( 1 - 398. iT - 7.95e4T^{2} \)
47 \( 1 + 176. iT - 1.03e5T^{2} \)
53 \( 1 - 131. iT - 1.48e5T^{2} \)
59 \( 1 - 654.T + 2.05e5T^{2} \)
61 \( 1 + 120.T + 2.26e5T^{2} \)
67 \( 1 + 310. iT - 3.00e5T^{2} \)
71 \( 1 + 400.T + 3.57e5T^{2} \)
73 \( 1 + 243. iT - 3.89e5T^{2} \)
79 \( 1 - 553.T + 4.93e5T^{2} \)
83 \( 1 + 756. iT - 5.71e5T^{2} \)
89 \( 1 + 80.1T + 7.04e5T^{2} \)
97 \( 1 + 1.10e3iT - 9.12e5T^{2} \)
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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

−8.291873611187854006842095537476, −7.60123775786496653545902087021, −6.91066186221949859099543521203, −6.20220851884038192561221006329, −5.19208010947653784634598832814, −4.45721565717215445250033683782, −3.22664717279421976539576011856, −2.49843957876396792451094506957, −1.18071549634752870099107060319, −0.37036535816253886462649550025, 1.21143135473428404046741468497, 2.31111213033764325647753014869, 3.47374845251050066080306907759, 4.07753635431654939132569353661, 5.18360490582439903557757144565, 5.78621672022298949313648431573, 6.66326967605597784807901054606, 7.62967474167803303135359313629, 8.411345360852994940456180229166, 9.115933993606922661109550739563

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