Properties

Degree 2
Conductor $ 3 \cdot 5 \cdot 7 $
Sign $0.245 + 0.969i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.63 − 2.63i)2-s + (−2.54 + 1.59i)3-s + 9.88i·4-s + (−4.15 + 2.78i)5-s + (10.8 + 2.49i)6-s + (2.39 − 6.57i)7-s + (15.5 − 15.5i)8-s + (3.91 − 8.10i)9-s + (18.2 + 3.60i)10-s + 5.66i·11-s + (−15.7 − 25.1i)12-s + (−1.68 + 1.68i)13-s + (−23.6 + 11.0i)14-s + (6.11 − 13.6i)15-s − 42.1·16-s + (14.0 − 14.0i)17-s + ⋯
L(s)  = 1  + (−1.31 − 1.31i)2-s + (−0.846 + 0.531i)3-s + 2.47i·4-s + (−0.830 + 0.556i)5-s + (1.81 + 0.415i)6-s + (0.342 − 0.939i)7-s + (1.93 − 1.93i)8-s + (0.434 − 0.900i)9-s + (1.82 + 0.360i)10-s + 0.514i·11-s + (−1.31 − 2.09i)12-s + (−0.129 + 0.129i)13-s + (−1.68 + 0.786i)14-s + (0.407 − 0.913i)15-s − 2.63·16-s + (0.824 − 0.824i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.245 + 0.969i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.245 + 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(105\)    =    \(3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $0.245 + 0.969i$
motivic weight  =  \(2\)
character  :  $\chi_{105} (83, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 105,\ (\ :1),\ 0.245 + 0.969i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(0.350300 - 0.272742i\)
\(L(\frac12)\)  \(\approx\)  \(0.350300 - 0.272742i\)
\(L(2)\)   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 \( 1 + (2.54 - 1.59i)T \)
5 \( 1 + (4.15 - 2.78i)T \)
7 \( 1 + (-2.39 + 6.57i)T \)
good2 \( 1 + (2.63 + 2.63i)T + 4iT^{2} \)
11 \( 1 - 5.66iT - 121T^{2} \)
13 \( 1 + (1.68 - 1.68i)T - 169iT^{2} \)
17 \( 1 + (-14.0 + 14.0i)T - 289iT^{2} \)
19 \( 1 - 24.0T + 361T^{2} \)
23 \( 1 + (-3.17 + 3.17i)T - 529iT^{2} \)
29 \( 1 - 24.1T + 841T^{2} \)
31 \( 1 + 23.8iT - 961T^{2} \)
37 \( 1 + (13.8 - 13.8i)T - 1.36e3iT^{2} \)
41 \( 1 - 53.4T + 1.68e3T^{2} \)
43 \( 1 + (-25.9 - 25.9i)T + 1.84e3iT^{2} \)
47 \( 1 + (27.3 - 27.3i)T - 2.20e3iT^{2} \)
53 \( 1 + (-22.4 + 22.4i)T - 2.80e3iT^{2} \)
59 \( 1 + 14.2iT - 3.48e3T^{2} \)
61 \( 1 + 90.2iT - 3.72e3T^{2} \)
67 \( 1 + (0.492 - 0.492i)T - 4.48e3iT^{2} \)
71 \( 1 - 54.2iT - 5.04e3T^{2} \)
73 \( 1 + (-30.4 + 30.4i)T - 5.32e3iT^{2} \)
79 \( 1 - 58.3iT - 6.24e3T^{2} \)
83 \( 1 + (55.7 + 55.7i)T + 6.88e3iT^{2} \)
89 \( 1 + 109. iT - 7.92e3T^{2} \)
97 \( 1 + (-48.0 - 48.0i)T + 9.40e3iT^{2} \)
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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

−12.59538599577073352520320344407, −11.67527827794582494697861257243, −11.17549493821871397750085564080, −10.19341226328861778150796894336, −9.541201361461283126707936863591, −7.87235220335608423116553122143, −7.09173414238218928048488613503, −4.49286363575771888528872802967, −3.25821979495518989586981155858, −0.77775461455980263043742307130, 1.06356229310377854849745520878, 5.15409748263365842574253466065, 5.91432773167995340584896691671, 7.31381039993418809704504427958, 8.097677675852507839617789604205, 8.988973065409706549159498576659, 10.38419512741396790474593597429, 11.52374837650546466915053786436, 12.46351251464397968010299400556, 14.09866395112936297346625240814

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