Properties

Degree 2
Conductor $ 3 \cdot 5 \cdot 7 $
Sign $-0.518 + 0.855i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.28 − 2.28i)2-s + (−2.80 + 1.07i)3-s − 6.40i·4-s + (1.80 − 4.66i)5-s + (−3.93 + 8.83i)6-s + (−3.20 − 6.22i)7-s + (−5.48 − 5.48i)8-s + (6.69 − 6.01i)9-s + (−6.50 − 14.7i)10-s + 11.1i·11-s + (6.88 + 17.9i)12-s + (−5.82 − 5.82i)13-s + (−21.5 − 6.86i)14-s + (−0.0592 + 14.9i)15-s + 0.592·16-s + (6.84 + 6.84i)17-s + ⋯
L(s)  = 1  + (1.14 − 1.14i)2-s + (−0.933 + 0.358i)3-s − 1.60i·4-s + (0.361 − 0.932i)5-s + (−0.656 + 1.47i)6-s + (−0.458 − 0.888i)7-s + (−0.685 − 0.685i)8-s + (0.743 − 0.668i)9-s + (−0.650 − 1.47i)10-s + 1.01i·11-s + (0.573 + 1.49i)12-s + (−0.448 − 0.448i)13-s + (−1.53 − 0.490i)14-s + (−0.00395 + 0.999i)15-s + 0.0370·16-s + (0.402 + 0.402i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.518 + 0.855i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.518 + 0.855i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(105\)    =    \(3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $-0.518 + 0.855i$
motivic weight  =  \(2\)
character  :  $\chi_{105} (62, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 105,\ (\ :1),\ -0.518 + 0.855i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(0.892872 - 1.58527i\)
\(L(\frac12)\)  \(\approx\)  \(0.892872 - 1.58527i\)
\(L(2)\)   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 \( 1 + (2.80 - 1.07i)T \)
5 \( 1 + (-1.80 + 4.66i)T \)
7 \( 1 + (3.20 + 6.22i)T \)
good2 \( 1 + (-2.28 + 2.28i)T - 4iT^{2} \)
11 \( 1 - 11.1iT - 121T^{2} \)
13 \( 1 + (5.82 + 5.82i)T + 169iT^{2} \)
17 \( 1 + (-6.84 - 6.84i)T + 289iT^{2} \)
19 \( 1 - 25.0T + 361T^{2} \)
23 \( 1 + (-23.3 - 23.3i)T + 529iT^{2} \)
29 \( 1 + 10.6T + 841T^{2} \)
31 \( 1 - 26.9iT - 961T^{2} \)
37 \( 1 + (20.8 + 20.8i)T + 1.36e3iT^{2} \)
41 \( 1 - 32.9T + 1.68e3T^{2} \)
43 \( 1 + (-1.25 + 1.25i)T - 1.84e3iT^{2} \)
47 \( 1 + (59.1 + 59.1i)T + 2.20e3iT^{2} \)
53 \( 1 + (26.0 + 26.0i)T + 2.80e3iT^{2} \)
59 \( 1 - 70.2iT - 3.48e3T^{2} \)
61 \( 1 - 14.1iT - 3.72e3T^{2} \)
67 \( 1 + (6.14 + 6.14i)T + 4.48e3iT^{2} \)
71 \( 1 - 39.0iT - 5.04e3T^{2} \)
73 \( 1 + (-51.1 - 51.1i)T + 5.32e3iT^{2} \)
79 \( 1 - 16.8iT - 6.24e3T^{2} \)
83 \( 1 + (31.3 - 31.3i)T - 6.88e3iT^{2} \)
89 \( 1 - 70.8iT - 7.92e3T^{2} \)
97 \( 1 + (-114. + 114. i)T - 9.40e3iT^{2} \)
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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

−12.83413089000031147749706164714, −12.33710537308209737054617657216, −11.29670027783871255120845644897, −10.15192238139616385647876259583, −9.623745271630045629072213015728, −7.25135352754120466003854675731, −5.53748296953441699360626929374, −4.81928401238143645654350010562, −3.60540697112723841483763225422, −1.25450594003756216512570658190, 3.08348372598808906320717482914, 5.04295222574483047241547681011, 5.97016142738505521181432636956, 6.66679091221384130521439151817, 7.72351018501950388059793211039, 9.593989020488294257871834722575, 11.09254316654064445467818509358, 12.04673386015024795102266974361, 13.06351787850773521761892283955, 13.93850753994870002147174417294

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