Properties

Degree 2
Conductor $ 2^{2} \cdot 3 \cdot 5^{2} \cdot 7 $
Sign $0.894 + 0.447i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  i·3-s i·7-s − 9-s + 11-s + 4i·13-s − 2i·17-s + 4·19-s − 21-s + 7i·23-s + i·27-s + 9·29-s − 2·31-s i·33-s i·37-s + 4·39-s + ⋯
L(s)  = 1  − 0.577i·3-s − 0.377i·7-s − 0.333·9-s + 0.301·11-s + 1.10i·13-s − 0.485i·17-s + 0.917·19-s − 0.218·21-s + 1.45i·23-s + 0.192i·27-s + 1.67·29-s − 0.359·31-s − 0.174i·33-s − 0.164i·37-s + 0.640·39-s + ⋯

Functional equation

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

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(2100\)    =    \(2^{2} \cdot 3 \cdot 5^{2} \cdot 7\)
\( \varepsilon \)  =  $0.894 + 0.447i$
motivic weight  =  \(1\)
character  :  $\chi_{2100} (1849, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 2100,\ (\ :1/2),\ 0.894 + 0.447i)\)
\(L(1)\)  \(\approx\)  \(1.812408588\)
\(L(\frac12)\)  \(\approx\)  \(1.812408588\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;5,\;7\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{2,\;3,\;5,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + iT \)
5 \( 1 \)
7 \( 1 + iT \)
good11 \( 1 - T + 11T^{2} \)
13 \( 1 - 4iT - 13T^{2} \)
17 \( 1 + 2iT - 17T^{2} \)
19 \( 1 - 4T + 19T^{2} \)
23 \( 1 - 7iT - 23T^{2} \)
29 \( 1 - 9T + 29T^{2} \)
31 \( 1 + 2T + 31T^{2} \)
37 \( 1 + iT - 37T^{2} \)
41 \( 1 - 8T + 41T^{2} \)
43 \( 1 + 9iT - 43T^{2} \)
47 \( 1 + 4iT - 47T^{2} \)
53 \( 1 - 6iT - 53T^{2} \)
59 \( 1 + 4T + 59T^{2} \)
61 \( 1 - 4T + 61T^{2} \)
67 \( 1 + 9iT - 67T^{2} \)
71 \( 1 - 5T + 71T^{2} \)
73 \( 1 - 10iT - 73T^{2} \)
79 \( 1 - 15T + 79T^{2} \)
83 \( 1 + 6iT - 83T^{2} \)
89 \( 1 + 8T + 89T^{2} \)
97 \( 1 + 10iT - 97T^{2} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−9.146465870184735332934552115827, −8.188741486731160862230965141119, −7.32299928303650050548369993068, −6.90661490351405145622428119943, −5.96931559488246018620684285268, −5.08580509834219888984809235677, −4.10716289327802357148768155796, −3.16746886683307598405410565243, −1.99243152912656953148516874951, −0.926719901077701485248323616985, 0.907060684603556304485651327637, 2.52870192577483275079273184112, 3.27550462808842902712028244703, 4.35614797214547920585632639742, 5.10506954473409917021379668141, 5.97022906938966719531904234320, 6.66280959340880125256554989821, 7.86820580502641555615456375265, 8.351351856227891143231312056968, 9.220977692434289125813218020240

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