Properties

Degree 2
Conductor $ 3 \cdot 5 \cdot 7 $
Sign $-0.0585 + 0.998i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.897 − 1.55i)2-s + (2.08 − 2.15i)3-s + (0.387 + 0.671i)4-s + (−4.89 − 0.996i)5-s + (−1.48 − 5.17i)6-s + (1.39 − 6.85i)7-s + 8.57·8-s + (−0.316 − 8.99i)9-s + (−5.94 + 6.72i)10-s + (1.00 − 0.581i)11-s + (2.25 + 0.562i)12-s + 12.4i·13-s + (−9.41 − 8.32i)14-s + (−12.3 + 8.49i)15-s + (6.14 − 10.6i)16-s + (9.31 + 16.1i)17-s + ⋯
L(s)  = 1  + (0.448 − 0.777i)2-s + (0.694 − 0.719i)3-s + (0.0969 + 0.167i)4-s + (−0.979 − 0.199i)5-s + (−0.247 − 0.862i)6-s + (0.199 − 0.979i)7-s + 1.07·8-s + (−0.0351 − 0.999i)9-s + (−0.594 + 0.672i)10-s + (0.0916 − 0.0528i)11-s + (0.188 + 0.0468i)12-s + 0.959i·13-s + (−0.672 − 0.594i)14-s + (−0.824 + 0.566i)15-s + (0.384 − 0.665i)16-s + (0.547 + 0.949i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0585 + 0.998i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.0585 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(105\)    =    \(3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $-0.0585 + 0.998i$
motivic weight  =  \(2\)
character  :  $\chi_{105} (44, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 105,\ (\ :1),\ -0.0585 + 0.998i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(1.34797 - 1.42942i\)
\(L(\frac12)\)  \(\approx\)  \(1.34797 - 1.42942i\)
\(L(2)\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{3,\;5,\;7\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{3,\;5,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad3 \( 1 + (-2.08 + 2.15i)T \)
5 \( 1 + (4.89 + 0.996i)T \)
7 \( 1 + (-1.39 + 6.85i)T \)
good2 \( 1 + (-0.897 + 1.55i)T + (-2 - 3.46i)T^{2} \)
11 \( 1 + (-1.00 + 0.581i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 - 12.4iT - 169T^{2} \)
17 \( 1 + (-9.31 - 16.1i)T + (-144.5 + 250. i)T^{2} \)
19 \( 1 + (15.2 - 26.3i)T + (-180.5 - 312. i)T^{2} \)
23 \( 1 + (-13.7 + 23.7i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 - 52.6iT - 841T^{2} \)
31 \( 1 + (17.2 + 29.9i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 + (0.357 + 0.206i)T + (684.5 + 1.18e3i)T^{2} \)
41 \( 1 - 17.2iT - 1.68e3T^{2} \)
43 \( 1 + 7.86iT - 1.84e3T^{2} \)
47 \( 1 + (-17.4 + 30.1i)T + (-1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (17.8 + 30.9i)T + (-1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (32.3 - 18.6i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-25.4 + 44.0i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (24.9 - 14.4i)T + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + 66.8iT - 5.04e3T^{2} \)
73 \( 1 + (46.7 - 27.0i)T + (2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (16.6 - 28.8i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + 72.0T + 6.88e3T^{2} \)
89 \( 1 + (41.4 + 23.9i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + 66.7iT - 9.40e3T^{2} \)
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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

−12.89734021987186656576183918840, −12.44822238105462422377831646284, −11.38511740061884418190504913587, −10.40329042998787848299641716638, −8.610413578531985978646632280784, −7.73480812629887720393928482427, −6.79938857547869055919093499158, −4.25947161184069613824690285553, −3.48426548347892354142131728728, −1.57613036325070355046080731687, 2.86247769253518071093123935730, 4.52701342831929410060896003494, 5.53762427623963489106785086179, 7.22002765270707820872329999452, 8.139099093344100683597314126594, 9.316462674768534278822533342223, 10.67316394590233893100254518618, 11.57920869720018319270243457093, 13.05929284955312168682698132283, 14.18473174438198004403180721414

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